From 532ccab712d065433958d8d6d2a1640cf34228b9 Mon Sep 17 00:00:00 2001 From: Dan Fortunato Date: Tue, 28 Oct 2025 16:26:34 -0400 Subject: [PATCH 1/7] Implement triangle nodes and modes --- jacobi.m | 33 +++++++++ koornwinder.m | 66 +++++++++++++++++ trianglepts.m | 200 ++++++++++++++++++++++++++++++++++++++++++++++++++ trilattice.m | 24 ++++++ tuples.m | 56 ++++++++++++++ 5 files changed, 379 insertions(+) create mode 100644 jacobi.m create mode 100644 koornwinder.m create mode 100644 trianglepts.m create mode 100644 trilattice.m create mode 100644 tuples.m diff --git a/jacobi.m b/jacobi.m new file mode 100644 index 0000000..f826770 --- /dev/null +++ b/jacobi.m @@ -0,0 +1,33 @@ +function p = jacobi(n, a, b, x) +%JACOBI Evaluate Jacobi polynomials. +% P = JACOBI(N, A, B, X) evaluates the Jacobi polynomials P_N^{A,B} at the +% points X, for N >= 0, A, B > -1, and X in [-1, 1]. + +n = n(:); +x = x(:); + +if ( any(n < 0) ) + error('N must be greater than 0.'); +end + +if ( a <= -1 || b <= -1 ) + error('A and B must be greater than -1.'); +end + +nmax = max(n); +m = numel(x); +p = zeros(m, nmax+1); + +p(:,1) = 1; +if ( nmax < 1 ), return, end +p(:,2) = (1+(a+b)/2)*x + (a-b)/2; +for k = 2:nmax + c1 = 2*k*(k+a+b)*(2*k-2+a+b); + c2 = (2*k-1+a+b)*(2*k+a+b)*(2*k-2+a+b); + c3 = (2*k-1+a+b)*(a+b)*(a-b); + c4 = -2*(k-1+a)*(k-1+b)*(2*k+a+b); + p(:,k+1) = ((c3+c2*x).*p(:,k) + c4*p(:,k-1)) / c1; +end +p = p(:,n+1); + +end diff --git a/koornwinder.m b/koornwinder.m new file mode 100644 index 0000000..f487461 --- /dev/null +++ b/koornwinder.m @@ -0,0 +1,66 @@ +function [K, Kx, Ky] = koornwinder(n, x, y) +%KOORNWINDER Evaluate Koornwinder polynomials. +% K = KOORNWINDER(N, X, Y) constructs the Vandermonde matrix K mapping +% NPOLY = (N+1)(N+2)/2 coefficients of an order-N Koornwinder polynomial +% expansion to values of the expansion at the given points (X,Y). The points +% (X,Y) must lie in the reference triangle with vertices (0,0), (1,0), (0,1). +% If X and Y are M x 1 vectors, then K is an M x NPOLY matrix. +% +% [K, KX, KY] = KOORNWINDER(N, X, Y) also constructs Vandermonde matrices KX +% and KY for the x- and y-derivatives of the Koornwinder polynomials. +% +% The Koornwinder polynomials K_{n,k}(x,y) are defined as +% +% K_{n,k}(x,y) = P_{n-k}^(2k+1,0)(2x-1) * P_k(2y/(1-x)-1) * (1-x)^k +% +% for 0<=n<=N, 0<=k<=n, and (x,y) in the reference triangle. + +if ( nargin < 2 ) + [x, y] = trianglepts(n+1); +end + +x = x(:); +y = y(:); + +m = numel(x); +npoly = (n+1)*(n+2)/2; +K = zeros(m, npoly); +Kx = zeros(m, npoly); +Ky = zeros(m, npoly); + +% Compute the scaled Legendre polynomials P_k(2*(y/(1-x))-1) (1-x)^k +z = 2*y-(1-x); +leg = zeros(m, n+1); +legx = zeros(m, n+1); +legy = zeros(m, n+1); +leg(:,1) = 1; legx(:,1) = 0; legy(:,1) = 0; +leg(:,2) = z; legx(:,2) = 1; legy(:,2) = 2; +for k = 2:n + leg(:,k+1) = ((2*k-1)*z.*leg(:,k) - (k-1)*(1-x).^2.*leg(:,k-1)) / k; + legx(:,k+1) = ((2*k-1)*( leg(:,k) + z.*legx(:,k)) - (k-1)*((1-x).^2.*legx(:,k-1) - 2*(1-x).*leg(:,k-1))) / k; + legy(:,k+1) = ((2*k-1)*(2*leg(:,k) + z.*legy(:,k)) - (k-1)*(1-x).^2.*legy(:,k-1)) / k; +end + +i = 1; +for nn = 0:n + for kk = 0:nn + scl = sqrt(1/(2*(2*kk+1)*(nn+1))); + jac = jacobi(nn-kk, 2*kk+1, 0, 2*x-1); + jacx = 2*djacobi(nn-kk, 2*kk+1, 0, 2*x-1); + K(:,i) = leg(:,kk+1) .*jac/scl; + Ky(:,i) = legy(:,kk+1).*jac/scl; + Kx(:,i) = legx(:,kk+1).*jac/scl + leg(:,kk+1).*jacx/scl; + i = i+1; + end +end + +end + +function djac = djacobi(n, a, b, x) + +djac = zeros(size(x)); +if ( n > 0 ) + djac = 0.5*(n+a+b+1)*jacobi(n-1,a+1,b+1,x); +end + +end diff --git a/trianglepts.m b/trianglepts.m new file mode 100644 index 0000000..fbdc003 --- /dev/null +++ b/trianglepts.m @@ -0,0 +1,200 @@ +function [x, y, w, t2v] = trianglepts(n, opts) +%TRIANGLEPTS Generate nodes on the triangle. +% [X, Y] = TRIANGLEPTS(N) generates an order-N set of nodes (X,Y) in the +% (0,0)-(1,0)-(0,1) unit triangle using Isaac's recursive algorithm [1]. X and +% Y are vectors of length N*(N+1)/2. By default, the starting 1D node set is +% taken to be the second-kind Chebyshev nodes (i.e., Chebyshev nodes with +% endpoints). The resulting nodes (X,Y) are fully symmetric in the triangle +% and match the starting 1D node set on the edges of the triangle. +% +% [X, Y] = TRIANGLEPTS(..., type='cheb2') is equivalent to the above. Other +% possible values for the node type include: +% +% First-kind Chebyshev: 'cheb1' 'gc' 'chebyshev1' +% Second-kind Chebyshev: 'cheb2' 'lgc' 'chebyshev2' 'cheb' +% Gauss-Legendre: 'leg' 'gl' 'legendre' +% Gauss-Lobatto: 'lob' 'lgl' 'lobatto' +% Equally spaced nodes: 'equi' 'uni' 'lin' 'equispaced' 'uniform' 'linspace' +% +% [X, Y, W] = TRIANGLEPTS(N) also returns a vector of order-N interpolatory +% quadrature weights W for integration on the unit triangle. +% +% [X, Y, W, T2V] = TRIANGLEPTS(N) also returns the (N-1)^2 x 3 matrix T2V +% encoding triangle-to-vertex connectivity of the Delaunay triangulation of +% the nodes (X,Y). The vertices of the i-th triangle are the elements of X and +% Y with indices given by T2V(i,:). +% +% [X, Y, ~, T2V] = TRIANGLEPTS(..., weights=false) skips computation of the +% quadrature weights W. +% +% References: +% +% [1] Tobin Isaac, "Recursive, parameter-free, explicitly defined +% interpolation nodes for simplices", SIAM J. Sci. Comput., 42 (2020), +% pp. A4046-A4062. + +arguments + n + opts.dim = 2 + opts.type = 'cheb2' + opts.domain = 'unit' + opts.weights {mustBeNumericOrLogical} = true +end + +switch lower(opts.type) + case {'leg', 'legendre', 'gl'} + family = @(n) legpts(n, [0 1]); + case {'lob', 'lobatto', 'lgl'} + family = @(n) lobpts_(n, [0 1]); + case {'cheb1', 'gc', 'chebyshev1'} + family = @(n) chebpts(n, [0 1], 1); + case {'cheb2', 'lgc', 'chebyshev2', 'cheb'} + family = @(n) chebpts(n, [0 1], 2); + case {'equi', 'equispaced', 'uni', 'uniform', 'lin', 'linspace'} + family = @(n) linspace(0, 1, n).'; + otherwise + error('Unknown node set.'); +end + +dim = opts.dim; + +% The number of polynomials up to degree n in dim dimensions +N = nchoosek(n+dim-1, dim); +x = zeros(N, dim+1); +i = 1; +for idx = tuples(dim+1, n-1) + x(i,:) = recursive(dim+1, n-1, idx+1, family); + i = i+1; +end +x = from_unit(to_unit(x, 'barycentric'), opts.domain); + +y = x(:,2); +x = x(:,1); + +w = []; +if ( nargout > 2 && opts.weights ) + % Compute interpolatory quadrature weights + K = koornwinder(n-1, x, y); + g = zeros(1, size(K,2)); + g(1) = sqrt(2)/2; + w = g / K; + w = w(:); +end + +if ( nargout > 3 ) + % Compute triangle-to-vertex connectivity + t2v = trilattice(n); +end + +end + +function [x, w, v] = lobpts_(n, dom) + +if ( n == 1 ) + x = 0; + w = 2; + v = 1; +else + [x, w, v] = lobpts(n); +end +x = (x+1)/2*diff(dom) + dom(1); +w = w/2*diff(dom); + +end + +function b = recursive(d, n, alpha, family) +% The barycentric d-simplex coordinates for a multiindex alpha with length n, +% based on a 1D node family. + +xn = family(n+1); +b = zeros(d, 1); +if ( isempty(xn) ) + return +end + +if ( d == 2 ) + b = xn([alpha(1) ; alpha(2)]); + return +end + +weight = 0; +for i = 1:d + alpha_noti = alpha([1:i-1 i+1:end]); + n_noti = n - (alpha(i)-1); + w = xn(n_noti+1); + br = recursive(d-1, n_noti, alpha_noti, family); + b(1:i-1) = b(1:i-1) + w * br(1:i-1); + b(i+1:end) = b(i+1:end) + w * br(i:end); + weight = weight + w; +end + +b = b / weight; + +end + +function y = from_unit(x, domain) + +switch lower(domain) + case 'unit' + y = x; + case 'biunit' + y = 2*x-1; + case 'barycentric' + z = 1 - sum(x, 2); + y = [x z]; + case 'equilateral' + d = size(x, 2); + y = x - 1/(d+1); % Shift centroid to zero + y = unit_to_equilateral(d, y); + y = 2*y; % Scale edge length to 2 + otherwise + error('Not implemented.') +end + +end + +function x = unit_to_equilateral(d, x) + +if ( d > 1 ) + % Move the top vertex over the centroid + x(:,1:d-1) = x(:,1:d-1) + x(:,d)/d; + % Make the projection onto the lesser dimensions equilateral + x(:,1:d-1) = unit_to_equilateral(d-1, x(:,1:d-1)); + % Scale the vertical dimension + x(:,d) = x(:,d) * sqrt((d+1)/(2*d)); +end + +end + +function y = to_unit(x, domain) + +switch lower(domain) + case 'unit' + y = x; + case 'biunit' + y = (x+1)/2; + case 'barycentric' + y = x(:,1:end-1); + case 'equilateral' + d = size(x, 2); + y = x / 2; % Scale edge length to 1 + y = equilateral_to_unit(d, y); + y = y + 1/(d+1); % Shift the first vertex to zero + otherwise + error('Not implemented.'); +end + +end + +function x = equilateral_to_unit(d, x) + +if ( d > 1 ) + % Scale the vertical dimension + x(:,d) = x(:,d) / sqrt((d+1)/(2*d)); + % Make the projection onto the lesser dimensions right-angled + x(:,1:d-1) = equilateral_to_unit(d-1, x(:,1:d-1)); + % Move the top vertex over first vertex + x(:,1:d-1) = x(:,1:d-1) - x(:,d)/d; +end + +end diff --git a/trilattice.m b/trilattice.m new file mode 100644 index 0000000..37f01d0 --- /dev/null +++ b/trilattice.m @@ -0,0 +1,24 @@ +function T = trilattice(n) +%TRILATTICE Indices for a regular lattice of triangles. +% T = TRILATTICE(N) constructs an (N-1)^2 x 3 matrix T encoding triangle-to- +% vertex connectivity of the Delaunay triangulation of a regular lattice of +% nodes inside the unit triangle. The vertices of the i-th subtriangle have +% indices given by T(i,:). + +% Compute triangle-to-vertex connectivity +ntri = (n-1)^2; +T = zeros(ntri, 3); +k = 1; +colstart = 1; +for i = 0:n-2 + % Add triangles spanning columns i and i+1 + h = n-i-1; + ss = (colstart+1:colstart+h-1).'; + T(k,:) = [colstart colstart+1 colstart+1+h]; + T(k+1:k+h-1,:) = [ss ss+h ss+h+1]; + T(k+h:k+2*h-2,:) = [ss ss+1 ss+h+1]; + k = k+2*h-1; + colstart = colstart+h+1; +end + +end diff --git a/tuples.m b/tuples.m new file mode 100644 index 0000000..5271caa --- /dev/null +++ b/tuples.m @@ -0,0 +1,56 @@ +function idx = tuples(dim, n, type) +%TUPLES Generate all tuples with a given sum. +% IDX = TUPLES(DIM, N, '=') constructs a matrix IDX of all unique DIM-tuples +% whose entries sum to N, in lexicographic order. IDX(:,j) is a vector of +% length DIM. TUPLES(DIM, N) is equivalent to TUPLES(DIM, N, '='). +% +% IDX = TUPLES(DIM, N, '<=') constructs a matrix IDX of all unique DIM-tuples +% whose entries sum to at most N, in lexicographic order. IDX(:,j) is a vector +% of length DIM. +% +% TUPLES(DIM, N, '<') is equivalent to TUPLES(DIM, N-1, '<='). +% +% Examples: +% +% >> tuples(3, 2, '=') +% +% ans = +% +% 0 0 0 1 1 2 +% 0 1 2 0 1 0 +% 2 1 0 1 0 0 +% +% >> tuples(3, 2, '<=') +% +% ans = +% +% 0 0 0 0 0 0 1 1 1 2 +% 0 0 0 1 1 2 0 0 1 0 +% 0 1 2 0 1 0 0 1 0 0 + +arguments + dim + n + type = '=' +end + +switch type + case '=' + idx = []; + if ( dim <= 0 || n < 0 ), return, end + if ( dim == 1 ), idx = n; return, end + for k = 0:n-1 + idx1 = tuples(dim-1, n-k, '='); + idx = [idx [k*ones(1,size(idx1,2)) ; idx1]]; %#ok + end + idx = [idx [n ; zeros(dim-1, 1)]]; + case '<=' + idx = tuples(dim+1, n, '='); + idx = idx(1:dim,:); + case '<' + idx = tuples(dim, n-1, '<='); + otherwise + error('Type must be either ''='' or ''<=''.'); +end + +end From 298cf8c94546eccee5f4512ccc5486686d16d727 Mon Sep 17 00:00:00 2001 From: Dan Fortunato Date: Tue, 28 Oct 2025 17:52:43 -0400 Subject: [PATCH 2/7] Add triangle elements to surfacemesh --- +surfacemesh/+import/csv.m | 14 + +surfacemesh/+import/gmsh.m | 356 +++++++++++++++++++++++ +surfacemesh/patchtype.m | 6 + @surfacemesh/icosphere.m | 141 +++++++++ @surfacemesh/import.m | 22 +- @surfacemesh/mesh.m | 66 ++++- @surfacemesh/order.m | 24 +- @surfacemesh/private/buildConnectivity.m | 205 +++++++++++++ @surfacemesh/private/computeNormals.m | 28 +- @surfacemesh/quadwts.m | 24 +- @surfacemesh/resample.m | 36 ++- @surfacemesh/size.m | 22 ++ @surfacemesh/surfacemesh.m | 57 +++- @surfacemesh/volume.m | 6 +- @surfacemesh/wireframe.m | 110 +++++-- 15 files changed, 1035 insertions(+), 82 deletions(-) create mode 100644 +surfacemesh/+import/csv.m create mode 100644 +surfacemesh/+import/gmsh.m create mode 100644 +surfacemesh/patchtype.m create mode 100644 @surfacemesh/icosphere.m create mode 100644 @surfacemesh/size.m diff --git a/+surfacemesh/+import/csv.m b/+surfacemesh/+import/csv.m new file mode 100644 index 0000000..f0d7c87 --- /dev/null +++ b/+surfacemesh/+import/csv.m @@ -0,0 +1,14 @@ +function dom = csv(filename, n) + +if ( nargin < 2 ) + n = 8; +end + +A = readmatrix(filename); +nelem = size(A,1)/n^2; +x = mat2cell(reshape(A(:,1), n, []), n, n*ones(nelem,1)).'; +y = mat2cell(reshape(A(:,2), n, []), n, n*ones(nelem,1)).'; +z = mat2cell(reshape(A(:,3), n, []), n, n*ones(nelem,1)).'; +dom = surfacemesh(x, y, z); + +end diff --git a/+surfacemesh/+import/gmsh.m b/+surfacemesh/+import/gmsh.m new file mode 100644 index 0000000..8409035 --- /dev/null +++ b/+surfacemesh/+import/gmsh.m @@ -0,0 +1,356 @@ +function dom = gmsh(file) + +fid = fopen(file); + +line = strip(fgetl(fid)); +while ( line == "$Comments" ) + jump_to_end_block(fid, 'Comments'); + line = strip(fgetl(fid)); +end + +% Read mesh format +if ( line ~= "$MeshFormat" ) + error('Invalid Gmsh file.'); +end +format = split(strip(fgetl(fid))); +if ( length(format) ~= 3 ) + error('Invalid Gmsh file.'); +end +version = format{1}; +if ( ~any(format{2} == '01') ) + error('Invalid Gmsh file.'); +end +isascii = ( format{2} == '0' ); +if ( ~isascii ) + error('Binary Gmsh files are unsupported.'); +end +data_size = str2double(format{3}); +jump_to_end_block(fid, 'MeshFormat'); + +while true + line = skip_blank_lines(fid); + if ( feof(fid) ) + break + end + + if ( length(line) < 2 || line(1) ~= '$' ) + error('Invalid Gmsh file'); + end + + section = strip(line(2:end)); + + switch ( section ) + case 'Nodes' + [nodes, node_tags, node_entities] = read_nodes(fid); + case 'Elements' + [triangles, quads] = read_elements(fid, node_tags); + otherwise + jump_to_end_block(fid, section); + end +end + +fclose(fid); + +if ( ~isempty(triangles) && ~isempty(quads) ) + error('Meshes with both triangles and quads are not currently supported.'); +end + +if ( ~isempty(triangles) ) + npts = length(triangles{1}); + if ( ~all(cellfun(@length, triangles) == npts) ) + error('Meshes with nonuniform polynomial order are not currently supported.'); + end + n = (sqrt(8*npts+1)-1) / 2; + ii = gmsh2trianglepts(n); + V = equi2tri(n); + x = cell(length(triangles), 1); + y = cell(length(triangles), 1); + z = cell(length(triangles), 1); + for k = 1:length(triangles) + xyz = nodes(:,triangles{k}); + xyz = xyz(:,ii); + x{k} = V * xyz(1,:).'; + y{k} = V * xyz(2,:).'; + z{k} = V * xyz(3,:).'; + end + dom = surfacemesh(x, y, z, 'tri'); +elseif ( ~isempty(quads) ) + npts = length(quads{1}); + if ( ~all(cellfun(@length, quads) == npts) ) + error('Meshes with nonuniform polynomial order are not currently supported.'); + end + n = sqrt(npts); + ii = gmsh2quadpts(n); + V = equi2cheb(n); + x = cell(length(quads), 1); + y = cell(length(quads), 1); + z = cell(length(quads), 1); + for k = 1:length(quads) + xyz = nodes(:,quads{k}); + xyz = xyz(:,ii); + x{k} = V * reshape(xyz(1,:), n, n) * V.'; + y{k} = V * reshape(xyz(2,:), n, n) * V.'; + z{k} = V * reshape(xyz(3,:), n, n) * V.'; + end + dom = surfacemesh(x, y, z, 'quad'); +end + +end + +function [nodes, node_tags, node_entities] = read_nodes(fid) + +% numEntityBlocks(size_t) numNodes(size_t) minNodeTag(size_t) maxNodeTag(size_t) +line = fscanf(fid, '%d', 4); +[numEntityBlocks, numNodes, minNodeTag, maxNodeTag] = dealm(line); + +nodes = zeros(3, numNodes); +node_tags = zeros(1, numNodes); +node_entities = zeros(2, numNodes); + +idx = 0; +for k = 1:numEntityBlocks + % entityDim(int) entityTag(int) parametric(int; 0 or 1) numNodes(size_t) + line = fscanf(fid, '%d', 4); + [entityDim, entityTag, parametric, numNodesInBlock] = dealm(line); + if ( parametric ~= 0 ) + error('Parametric nodes are not implemented.'); + end + ixx = idx + (1:numNodesInBlock); + node_tags(ixx) = fscanf(fid, '%d', numNodesInBlock); + nodes(:,ixx) = reshape(fscanf(fid, '%g', 3*numNodesInBlock), 3, numNodesInBlock); + node_entities(1,ixx) = entityDim; + node_entities(2,ixx) = entityTag; + idx = idx + numNodesInBlock; +end + +jump_to_end_block(fid, 'Nodes'); + +end + +function [triangles, quads] = read_elements(fid, node_tags) + +element_types = dictionary; +element_types(15) = {{'vertex', 1}}; +element_types(1) = {{'line', 2}}; +element_types(8) = {{'line', 3}}; +element_types(26) = {{'line', 4}}; +element_types(27) = {{'line', 5}}; +element_types(28) = {{'line', 6}}; +element_types(62) = {{'line', 7}}; +element_types(63) = {{'line', 8}}; +element_types(64) = {{'line', 9}}; +element_types(65) = {{'line', 10}}; +element_types(66) = {{'line', 11}}; +element_types(2) = {{'triangle', 3}}; +element_types(9) = {{'triangle', 6}}; +element_types(20) = {{'triangle', 9}}; +element_types(21) = {{'triangle', 10}}; +element_types(22) = {{'triangle', 12}}; +element_types(23) = {{'triangle', 15}}; +element_types(25) = {{'triangle', 21}}; +element_types(42) = {{'triangle', 28}}; +element_types(43) = {{'triangle', 36}}; +element_types(44) = {{'triangle', 45}}; +element_types(45) = {{'triangle', 55}}; +element_types(46) = {{'triangle', 66}}; +element_types(3) = {{'quad', 4}}; +element_types(10) = {{'quad', 9}}; +element_types(16) = {{'quad', 8}}; +element_types(36) = {{'quad', 16}}; +element_types(37) = {{'quad', 25}}; +element_types(38) = {{'quad', 36}}; +element_types(39) = {{'quad', 12}}; +element_types(41) = {{'quad', 20}}; +element_types(47) = {{'quad', 49}}; +element_types(48) = {{'quad', 64}}; +element_types(49) = {{'quad', 81}}; +element_types(50) = {{'quad', 100}}; +element_types(51) = {{'quad', 121}}; + +inv_tags = zeros(max(node_tags), 1); +inv_tags(node_tags) = 1:length(node_tags); + +% numEntityBlocks(size_t) numElements(size_t) minElementTag(size_t) maxElementTag(size_t) +line = fscanf(fid, '%d', 4); +[numEntityBlocks, numElements, minElementTag, maxElementTag] = dealm(line); + +triangles = {}; +quads = {}; + +for k = 1:numEntityBlocks + % entityDim(int) entityTag(int) elementType(int) numElementsInBlock(size_t) + line = fscanf(fid, '%d', 4); + [entityDim, entityTag, elementType, numElementsInBlock] = dealm(line); + if ( ~isKey(element_types, elementType) ) + error('Unsupported element type.'); + end + elem = element_types{elementType}; + [elemShape, numNodesPerElem] = elem{:}; + % d = fromfile(f, c_size_t, int(num_ele * (1 + num_nodes_per_ele))).reshape((num_ele, -1)) + for j = 1:numElementsInBlock + line = fscanf(fid, '%d', numNodesPerElem+1); + elementTag = line(1); + nodes = line(2:end); + if ( elemShape == "triangle" ) + triangles{end+1} = inv_tags(nodes); %#ok<*AGROW> + elseif ( elemShape == "quad" ) + quads{end+1} = inv_tags(nodes); + end + end + +end + +end + +function ii = gmsh2trianglepts(n) + +ii = zeros(n); +ox = 1; +oy = n; +idx = 1; +nshell = ceil(n/3); +for shell = 1:nshell + ii(oy,ox) = idx; idx = idx+1; + if ( idx > n*(n+1)/2 ), break, end + + h = n-3*shell+2; + ii(oy-h,ox) = idx; idx = idx+1; + ii(oy,ox+h) = idx; idx = idx+1; + if ( idx > n*(n+1)/2 ), break, end + + o = n*(ox-1)+oy; + hypot = (o+3*shell-1):(n+1):(o+n*h-1); + ii(oy-(1:h-1),ox) = idx + (0:h-2); idx = idx + h-1; + ii(hypot) = idx + (0:h-2); idx = idx + h-1; + ii(oy,ox+(h-1:-1:1)) = idx + (0:h-2); idx = idx + h-1; + + ox = ox+1; + oy = oy-1; +end + +ii = flipud(ii); +ii = ii(:); +ii(ii == 0) = []; + +end + +function ii = gmsh2quadpts(n) + +ii = zeros(n); +ox = n; +oy = n; +idx = 1; +nshell = ceil(n/2); +for shell = 1:nshell + ii(oy,ox) = idx; idx = idx+1; + if ( idx > n^2 ), break, end + + h = n-2*shell; + ii(oy,ox-h-1) = idx; idx = idx+1; + ii(oy-h-1,ox-h-1) = idx; idx = idx+1; + ii(oy-h-1,ox) = idx; idx = idx+1; + if ( idx > n^2 ), break, end + + ii(oy,ox-(1:h)) = idx + (0:h-1); idx = idx + h; + ii(oy-(1:h),ox-h-1) = idx + (0:h-1); idx = idx + h; + ii(oy-h-1,ox-(h:-1:1)) = idx + (0:h-1); idx = idx + h; + ii(oy-(h:-1:1),ox) = idx + (0:h-1); idx = idx + h; + + ox = ox-1; + oy = oy-1; +end + +ii = flipud(ii); +ii = ii(:); + +end + +function V = equi2tri(n) + [x, y] = trianglepts(n); + [xgmsh, ygmsh] = trianglepts(n, type='linspace'); + K = koornwinder(n-1, x, y); + Kgmsh = koornwinder(n-1, xgmsh, ygmsh); + V = K / Kgmsh; +end + +function V = equi2cheb(n) + xgmsh = linspace(-1, 1, n).'; + cheb2equi = bary(xgmsh, eye(n)); + V = inv(cheb2equi); +end + +function jump_to_end_block(fid, block) + found = false; + while ~feof(fid) + line = fgetl(fid); + if strip(line) == sprintf("$End%s", block) + found = true; + break + end + end + if ( ~found ) + error('$%s not closed by $End%s.', block); + end +end + +function line = skip_blank_lines(fid) + while ~feof(fid) + line = strip(fgetl(fid)); + if ~isempty(line) + break + end + end +end + +function varargout = dealm(x) + n = size(x, 2); + if ( n == 1 ) + n = size(x, 1); + x = x(:).'; + end + if ( nargout > n ) + error('Too many outputs.'); + end + varargout = cell(1, nargout); + for k = 1:nargout + varargout{k} = x(:,k); + end +end + +function y = quadTensorProductOrder(x, n) + +y = zeros(n); + +y(1) = x(1); + +if ( n > 1 ) + y(n) = x(2); + y(n^2) = x(3); + y(n*(n-1)+1) = x(4); +end + +for i = 1:n-2 + y(i+1) = x(4+i); +end + +for i = 1:n-2 + y(n^2-i) = x(4+2*(n-2)+i); +end + +% Interior (recursion) +if ( n > 2 ) + % Reorder interior nodes + interior = x((4+4*(n-2)+1):end); + interior = quadTensorProductOrder(interior, n-2); + + % Copy into full node list + for j = 1:n-2 + y(j*n+1) = x(4+3*(n-2)+n-2-j+1); + for i = 1:n-2 + y(j*n+i+1) = interior((j-1)*(n-2)+(i-1)+1); + end + y((j+1)*n) = x(4+(n-2)+j); + end +end + +end diff --git a/+surfacemesh/patchtype.m b/+surfacemesh/patchtype.m new file mode 100644 index 0000000..06001a2 --- /dev/null +++ b/+surfacemesh/patchtype.m @@ -0,0 +1,6 @@ +classdef patchtype < uint32 + enumeration + quad (0) + tri (1) + end +end diff --git a/@surfacemesh/icosphere.m b/@surfacemesh/icosphere.m new file mode 100644 index 0000000..d5c72c3 --- /dev/null +++ b/@surfacemesh/icosphere.m @@ -0,0 +1,141 @@ +function dom = icosphere(n, nref) +%ICOSPHERE Create an icosahedral sphere mesh. + +if ( nargin < 2 ) + nref = 0; +end + +% Create a regular unit icosahedron +[v, f] = icosahedron(); + +% Recursively subdivide triangle faces +for gen = 1:nref + nf = size(f, 1); + f_ = zeros(4*nf, 3); + % Loop over each triangle + for i = 1:nf + tri = f(i,:); + % Calculate mid points (add new points to v) + [a, v] = getMidPoint(tri(1), tri(2), v); + [b, v] = getMidPoint(tri(2), tri(3), v); + [c, v] = getMidPoint(tri(3), tri(1), v); + % Generate new subdivision triangles + nfc = [ tri(1) a c ; + tri(2) b a ; + tri(3) c b ; + a b c ]; + % Replace triangle with subdivision + idx = 4*(i-1)+1:4*i; + f_(idx,:) = nfc; + end + f = f_; +end + +% Remove duplicate vertices +[v, ~, ix] = unique(v, 'rows'); + +% Reassign faces to trimmed vertex list and remove any duplicate faces +f = unique(ix(f), 'rows'); + +dom0 = [0 0; 0 1; 1 0]; +dom0 = [dom0 zeros(3,1)]; +[uu0, vv0] = trianglepts(n, type='cheb2'); +uvw0 = [uu0 vv0 zeros(size(uu0)) ones(size(uu0))].'; + +nf = size(f, 1); +x = cell(nf, 1); +y = cell(nf, 1); +z = cell(nf, 1); +for k = 1:nf + tri = f(k,:); + dom = v(tri,:); + A = affine(dom0, dom); + xyz = A*uvw0; + nrm = sqrt(xyz(1,:).^2 + xyz(2,:).^2 + xyz(3,:).^2); + x{k} = xyz(1,:).' ./ nrm.'; + y{k} = xyz(2,:).' ./ nrm.'; + z{k} = xyz(3,:).' ./ nrm.'; +end + +dom = surfacemesh(x, y, z, 'tri'); + +end + +function A = affine(from, to) +%AFFINE Affine transformation. +% A = AFFINE(FROM, TO) + +if ( ~ismatrix(from) || ~ismatrix(to) || ~all(size(from) == size(to)) ) + error('Invalid points.'); +end + +n = size(from, 1); +dim = size(from, 2); + +w = ones(1, n); +A = [to.'; w] / [from.' ; w]; + +end + +function [i, v] = getMidPoint(t1, t2, v) +%GETMIDPOINT Calculate midpoint between two vertices. +% Calculate new vertex in sub-division and normalise to unit length +% then find or add it to v and return index. + +% Get vertex positions +p1 = v(t1,:); +p2 = v(t2,:); + +% Calculate mid point (on unit sphere) +pm = (p1 + p2) / 2; +pm = pm / norm(pm); + +% Add to vertices list, return index +i = size(v, 1) + 1; +v = [v ; pm]; + +end + +function [v,f] = icosahedron() +%ICOSAHEDRON Create a unit regular icosahedron. +% Returns 12 vertex and 20 face values. + +t = (1 + sqrt(5)) / 2; + +v = [ -1 t 0 ; + 1 t 0 ; + -1 -t 0 ; + 1 -t 0 ; + 0 -1 t ; + 0 1 t ; + 0 -1 -t ; + 0 1 -t ; + t 0 -1 ; + t 0 1 ; + -t 0 -1 ; + -t 0 1 ]; + +v = v / norm(v(1,:)); + +f = [ 1 12 6 ; + 1 6 2 ; + 1 2 8 ; + 1 8 11 ; + 1 11 12 ; + 2 6 10 ; + 6 12 5 ; + 12 11 3 ; + 11 8 7 ; + 8 2 9 ; + 4 10 5 ; + 4 5 3 ; + 4 3 7 ; + 4 7 9 ; + 4 9 10 ; + 5 10 6 ; + 3 5 12 ; + 7 3 11 ; + 9 7 8 ; + 10 9 2 ]; + +end diff --git a/@surfacemesh/import.m b/@surfacemesh/import.m index 7695c38..f29c971 100644 --- a/@surfacemesh/import.m +++ b/@surfacemesh/import.m @@ -1,11 +1,27 @@ function dom = import(filename, format, varargin) %IMPORT Import a surfacemesh. +if ( ~exist(filename, 'file') ) + error('File not found: ''%s''', filename); +end + +if ( nargin == 1 && (ischar(filename) || isstring(filename)) ) + [~, ~, ext] = fileparts(filename); + switch ext + case '.msh' + format = 'gmsh'; + case '.csv' + format = 'rhino'; + otherwise + error('Cannot determine mesh format from file extension.'); + end +end + switch lower(format) case 'gmsh' - dom = surfacemesh.fromGmsh(filename, varargin{:}); - case {'rhino', 'rhinoceros'} - dom = surfacemesh.fromRhino(filename, varargin{:}); + dom = surfacemesh.import.gmsh(filename, varargin{:}); + case {'csv', 'rhino', 'rhinoceros'} + dom = surfacemesh.import.csv(filename, varargin{:}); otherwise error('Unknown mesh format ''%s''.', format); end diff --git a/@surfacemesh/mesh.m b/@surfacemesh/mesh.m index 2f3fabe..980318b 100644 --- a/@surfacemesh/mesh.m +++ b/@surfacemesh/mesh.m @@ -1,30 +1,68 @@ function mesh(dom, varargin) %MESH Plot the mesh of a surface. +surfaceStyle = {'FaceColor', 'w', ... + 'AmbientStrength', 0.6, ... + 'DiffuseStrength', 0.4, ... + 'SpecularStrength', 0.3 }; + holdState = ishold(); x = dom.x; y = dom.y; z = dom.z; vn = dom.facenormals; +ne = length(dom); + +if ( ~all(dom.ptype == dom.ptype(1)) ) + error('Heterogeneous patch types are not yet supported.'); +end + +switch ( dom.ptype(1) ) + case 'tri' + n = order(dom)+1; + npts = length(dom.x{1}); + T = trilattice(n); + ntri = size(T, 1); + T_all = repmat(T, [1 1 ne]) + reshape((0:ne-1)*npts, 1, 1, []); + T_all = permute(T_all, [1 3 2]); + T_all = reshape(T_all, ntri*ne, 3); + x = [x{:}]; + y = [y{:}]; + z = [z{:}]; + x_all = x(:); + y_all = y(:); + z_all = z(:); + + hold on + patch('Faces', T_all, ... + 'Vertices', [x_all y_all z_all], ... + 'FaceVertexCData', 0*x_all, ... + surfaceStyle{:}, ... + varargin{:}); -for k = 1:length(dom) - surf(x{k}, y{k}, z{k}, 0*x{k}, 'FaceColor', 'None', varargin{:}) - hold on - % If the plot is not being added to another then plot the surface so - % that the lines are more easily discernable. - if ( ~holdState ) - % Plot the surface, making it slightly smaller so lines show up - % more clearly. - scl = 0.01; - surface(x{k} - scl*vn{k}(:,:,1), ... - y{k} - scl*vn{k}(:,:,2), ... - z{k} - scl*vn{k}(:,:,3), ... - 0*x{k}, 'FaceColor', 'w', 'EdgeColor', 'None'); - end + case 'quad' + for k = 1:length(dom) + surf(x{k}, y{k}, z{k}, 0*x{k}, 'FaceColor', 'None', varargin{:}) + hold on + % If the plot is not being added to another then plot the surface so + % that the lines are more easily discernable. + if ( ~holdState ) + % Plot the surface, making it slightly smaller so lines show up + % more clearly. + scl = 0.01; + surface(x{k} - scl*vn{k}(:,:,1), ... + y{k} - scl*vn{k}(:,:,2), ... + z{k} - scl*vn{k}(:,:,3), ... + 0*x{k}, ... + 'EdgeColor', 'none', ... + surfaceStyle{:}); + end + end end if ( ~holdState ) + view(3) axis equal hold off end diff --git a/@surfacemesh/order.m b/@surfacemesh/order.m index a86cb94..1f05b08 100644 --- a/@surfacemesh/order.m +++ b/@surfacemesh/order.m @@ -1,10 +1,26 @@ -function p = order(dom) +function [pu, pv] = order(dom) %ORDER Order of a SURFACEMESH. if ( isempty(dom) ) - p = []; -else - p = size(dom.x{1}, 1) - 1; + pu = []; + pv = []; + return +end + +if ( ~all(dom.ptype == dom.ptype(1)) ) + error('Heterogeneous patch types are not yet supported.'); +end + +switch ( dom.ptype(1) ) + case 'tri' + npts = length(dom.x{1}); + n = (sqrt(8*npts+1)-1) / 2; + pu = n-1; + pv = n-1; + case 'quad' + [nv, nu] = size(dom.x{1}); + pu = nu-1; + pv = nv-1; end end diff --git a/@surfacemesh/private/buildConnectivity.m b/@surfacemesh/private/buildConnectivity.m index 817b186..6e2a5e8 100644 --- a/@surfacemesh/private/buildConnectivity.m +++ b/@surfacemesh/private/buildConnectivity.m @@ -1,5 +1,210 @@ function connectivity = buildConnectivity(dom) +%connectivity = buildTriConnectivity(dom); +%connectivity = buildQuadConnectivity(dom); +connectivity = buildMixedConnectivity(dom); + +% switch ( dom.ptype ) +% case 'tri' +% connectivity = buildTriConnectivity(dom); +% case 'quad' +% connectivity = buildQuadConnectivity(dom); +% otherwise +% error('Unknown patch type.'); +% end + +end + +function connectivity = buildMixedConnectivity(dom) + +tol = 1e-12; +triIdx = find(dom.ptype == 'tri'); +quadIdx = find(dom.ptype == 'quad'); + +npts = length(dom.x{1}); +n = (sqrt(8*npts+1) - 1) / 2; +cidx = [1 n npts]; + +nodes = cell(length(dom), 1); +for k = triIdx(:).' + nodes{k} = [dom.x{k}(cidx) dom.y{k}(cidx) dom.z{k}(cidx)]; +end + +[nv, nu] = size(dom.x{1}); +%corners = [1 1; nv 1; nv nu; 1 nu]; +corners = [1 1; nv 1; 1 nu; nv nu]; +cidx = sub2ind([nv nu], corners(:,1), corners(:,2)); +cidx = cidx(:); + +for k = quadIdx(:).' + nodes{k} = [dom.x{k}(cidx) dom.y{k}(cidx) dom.z{k}(cidx)]; +end + +% Node j has (x,y,z) coordinate nodes(j,:) +nodes = cat(1, nodes{:}); +%[nodes, ~, elem2node] = unique(nodes, 'rows', 'stable'); +[nodes, ~, elem2node] = uniquetol(nodes, tol, 'ByRows', true); + +% Element i has corner nodes elem2node{i} +sz = zeros(length(dom), 1); +sz(triIdx) = 3; +sz(quadIdx) = 4; +elem2node = mat2cell(elem2node, sz, 1); + +% Node j touches elements node2elem{j} +node2elem = cell(length(nodes), 1); +for k = 1:length(dom) + for j = elem2node{k}(:).' + node2elem{j} = [node2elem{j} k]; + end +end + +% Edge j has nodes edges(j,:) +edges = cell(length(dom), 1); +for k = triIdx(:).' + ek = elem2node{k}; + edges{k} = ek([1 2 ; 2 3 ; 3 1]); +end + +for k = quadIdx(:).' + ek = elem2node{k}; + edges{k} = ek([1 2 ; 2 4 ; 4 3 ; 3 1]); +end + +edges = cat(1, edges{:}); +edges = sort(edges, 2); +%[edges, ~, elem2edge] = unique(edges, 'rows', 'stable'); +[edges, ~, elem2edge] = uniquetol(edges, tol, 'ByRows', true); + +% Element i has edges elem2edge{i} +elem2edge = mat2cell(elem2edge, sz, 1); + +% Edge j touches elements edge2elem{j} +edge2elem = cell(length(edges), 1); +for k = 1:length(dom) + for j = elem2edge{k}(:).' + edge2elem{j} = [edge2elem{j} k]; + end +end + +for j = 1:length(edge2elem) + if ( length(edge2elem{j}) < 2 ) + edge2elem{j} = [edge2elem{j} -1]; + end +end + +elem2elem = cell(length(dom), 1); + +% Element i has neighboring elements elem2elem(i,:) +%elem2elem = zeros(length(dom), 3); +%elem2elem_all = cell2mat(edge2elem(elem2edge)); +for k = 1:length(dom) + %idx = elem2elem_all(k,:) ~= k; + + nedges = length(elem2edge{k}); + for l = 1:nedges + idx = cat(3, edge2elem{elem2edge{k}}); + idx = idx(idx ~= k); + if ( any(idx) ) + elem2elem{k} = idx; %elem2elem_all(k, idx); + else + elem2elem{k} = -1; + end + end +end + +connectivity = struct(); +connectivity.nodes = nodes; +connectivity.elem2node = elem2node; +connectivity.node2elem = node2elem; +connectivity.edges = edges; +connectivity.elem2edge = elem2edge; +connectivity.edge2elem = edge2elem; +connectivity.elem2elem = elem2elem; + +end + +function connectivity = buildTriConnectivity(dom) + +npts = length(dom.x{1}); +n = (sqrt(8*npts+1) - 1) / 2; +cidx = [1 n npts]; + +% Node j has (x,y,z) coordinate nodes(j,:) +nodes = zeros(3*length(dom), 3); +j = 0; +for k = 1:length(dom) + nodes(j+(1:3),:) = [dom.x{k}(cidx) dom.y{k}(cidx) dom.z{k}(cidx)]; + j = j+3; +end +[nodes, ~, elem2node] = unique(nodes, 'rows', 'stable'); +%[nodes, ~, elem2node] = uniquetol(nodes, 'ByRows', true); + +% Element i has corner nodes elem2node(i,:) +elem2node = reshape(elem2node, 3, length(dom)).'; + +% Node j touches elements node2elem{j} +node2elem = cell(length(nodes), 1); +for k = 1:length(dom) + for j = elem2node(k,:) + node2elem{j} = [node2elem{j} k]; + end +end + +% Edge j has nodes edges(j,:) +edges = zeros(3*length(dom), 2); +j = 0; +for k = 1:length(dom) + edges(j+1,:) = elem2node(k, [1 2]); + edges(j+2,:) = elem2node(k, [2 3]); + edges(j+3,:) = elem2node(k, [3 1]); + j = j+3; +end +edges = sort(edges, 2); +[edges, ~, elem2edge] = unique(edges, 'rows', 'stable'); + +% Element i has edges elem2edge(i,:) +elem2edge = reshape(elem2edge, 3, length(dom)).'; + +% Edge j touches elements edge2elem{j} +edge2elem = cell(length(edges), 1); +for k = 1:length(dom) + for j = elem2edge(k,:) + edge2elem{j} = [edge2elem{j} k]; + end +end + +for j = 1:length(edge2elem) + if ( length(edge2elem{j}) < 2 ) + edge2elem{j} = [edge2elem{j} -1]; + end +end + +% Element i has neighboring elements elem2elem(i,:) +elem2elem = zeros(length(dom), 3); +elem2elem_all = cell2mat(edge2elem(elem2edge)); +for k = 1:length(dom) + idx = elem2elem_all(k,:) ~= k; + if ( any(idx) ) + elem2elem(k,:) = elem2elem_all(k, idx); + else + elem2elem(k,:) = -1; + end +end + +connectivity = struct(); +connectivity.nodes = nodes; +connectivity.elem2node = elem2node; +connectivity.node2elem = node2elem; +connectivity.edges = edges; +connectivity.elem2edge = elem2edge; +connectivity.edge2elem = edge2elem; +connectivity.elem2elem = elem2elem; + +end + +function connectivity = buildQuadConnectivity(dom) + [nv, nu] = size(dom.x{1}); %corners = [1 1; nv 1; nv nu; 1 nu]; corners = [1 1; nv 1; 1 nu; nv nu]; diff --git a/@surfacemesh/private/computeNormals.m b/@surfacemesh/private/computeNormals.m index 6636398..7855527 100644 --- a/@surfacemesh/private/computeNormals.m +++ b/@surfacemesh/private/computeNormals.m @@ -33,9 +33,16 @@ edges = dom.connectivity.edges; % Orient all normal vectors in the same direction -[nv, nu] = size(dom.x{1}); -%corners = [1 1; nv 1; nv nu; 1 nu]; -corners = [1 1; nv 1; 1 nu; nv nu]; +switch ( dom.ptype(1) ) + case 'tri' + npts = length(dom.x{1}); + n = (sqrt(8*npts+1)-1) / 2; + corners = [1 1; n 1; npts 1]; + case 'quad' + [nv, nu] = size(dom.x{1}); + %corners = [1 1; nv 1; nv nu; 1 nu]; + corners = [1 1; nv 1; 1 nu; nv nu]; +end processed = false(length(normal), 1); processed(1) = true; queue = zeros(length(dom), 1); queue(1) = 1; head = 1; tail = 2; @@ -43,13 +50,14 @@ % Pop from the queue k = queue(head); head = head+1; - xk = elem2node(k,:); + xk = elem2node{k}(:).'; % Find the matching point l = 1; - for j = elem2elem(k,:) + for iii = 1:length(elem2elem{k}) + j = elem2elem{k}(iii); if ( j > 0 && ~processed(j) ) - xj = elem2node(j,:); - xl = edges(elem2edge(k,l), 1); + xj = elem2node{j}(:).'; + xl = edges(elem2edge{k}(iii), 1); cj = corners(xj==xl,:); ck = corners(xk==xl,:); % If the normals are assumed to be smooth across patches, then @@ -71,14 +79,14 @@ % Now all vectors are pointing either inward or outward. % Make them point outward by checking the sign of the volume. -wu = chebtech2.quadwts(nu); wu = wu(:); -wv = chebtech2.quadwts(nv); wv = wv(:); + +W = quadwts(dom); V = 0; for k = 1:length(dom) I = (dom.x{k} .* normal{k}(:,:,1) + ... dom.y{k} .* normal{k}(:,:,2) + ... dom.z{k} .* normal{k}(:,:,3)) / 3; - V = V + sum(sum(((I .* wv) .* wu.') .* sqrt(dom.J{k}))); + V = V + sum(I .* W(:,:,k), 'all'); end if ( V < 0 ) diff --git a/@surfacemesh/quadwts.m b/@surfacemesh/quadwts.m index dd0ac5b..155ea5f 100644 --- a/@surfacemesh/quadwts.m +++ b/@surfacemesh/quadwts.m @@ -11,15 +11,29 @@ return end +if ( ~all(dom.ptype == dom.ptype(1)) ) + error('Heterogeneous patch types are not yet supported.'); +end + % If a patch uses an N x N discretization, then quadrature is performed on % that patch using N points. nelem = length(dom); -[nv, nu] = size(dom.x{1}); -wu = chebtech2.quadwts(nu); wu = wu(:); -wv = chebtech2.quadwts(nv); wv = wv(:); -W = zeros(nv, nu, nelem); +switch ( dom.ptype(1) ) + case 'tri' + npts = length(dom.x{1}); + n = (sqrt(8*npts+1)-1) / 2; + [~, ~, ww] = trianglepts(n); + W = zeros(npts, 1, nelem); + case 'quad' + [nv, nu] = size(dom.x{1}); + wu = chebtech2.quadwts(nu); wu = wu(:); + wv = chebtech2.quadwts(nv); wv = wv(:); + ww = wv .* wu.'; + W = zeros(nv, nu, nelem); +end + for k = 1:nelem - W(:,:,k) = wv .* wu.' .* sqrt(dom.J{k}); + W(:,:,k) = ww .* sqrt(dom.J{k}); end end diff --git a/@surfacemesh/resample.m b/@surfacemesh/resample.m index 27464f8..e3e35ed 100644 --- a/@surfacemesh/resample.m +++ b/@surfacemesh/resample.m @@ -4,18 +4,40 @@ return end -m = size(dom.x{1}, 1); -B = barymat(chebpts(n), chebpts(m)); +if ( ~all(dom.ptype == dom.ptype(1)) ) + error('Heterogeneous patch types are not yet supported.'); +end x = dom.x; y = dom.y; z = dom.z; -for k = 1:length(dom) - x{k} = B * x{k} * B.'; - y{k} = B * y{k} * B.'; - z{k} = B * z{k} * B.'; + +switch ( dom.ptype(1) ) + case 'tri' + m = order(dom)+1; + [xm, ym] = trianglepts(m); + [xn, yn] = trianglepts(n); + Km = koornwinder(m-1, xm, ym); + Knm = koornwinder(m-1, xn, yn); + B = Knm / Km; + for k = 1:length(dom) + x{k} = B * x{k}; + y{k} = B * y{k}; + z{k} = B * z{k}; + end + + case 'quad' + m = size(dom.x{1}, 1); + B = barymat(chebpts(n), chebpts(m)); + for k = 1:length(dom) + x{k} = B * x{k} * B.'; + y{k} = B * y{k} * B.'; + z{k} = B * z{k} * B.'; + end end -dom = surfacemesh(x, y, z); +ctvy = dom.connectivity; +dom = surfacemesh(x, y, z, dom.ptype); +dom.connectivity = ctvy; end diff --git a/@surfacemesh/size.m b/@surfacemesh/size.m new file mode 100644 index 0000000..ec542d3 --- /dev/null +++ b/@surfacemesh/size.m @@ -0,0 +1,22 @@ +function [nu, nv] = size(dom) +%SIZE Size of a SURFACEMESH. + +if ( isempty(dom) ) + nu = []; + nv = []; + return +end + +if ( ~all(dom.ptype == dom.ptype(1)) ) + error('Heterogeneous patch types are not yet supported.'); +end + +switch ( dom.ptype(1) ) + case 'tri' + nu = 1; + nv = length(dom.x{1}); + case 'quad' + [nv, nu] = size(dom.x{1}); +end + +end diff --git a/@surfacemesh/surfacemesh.m b/@surfacemesh/surfacemesh.m index eff3b70..7f009c2 100644 --- a/@surfacemesh/surfacemesh.m +++ b/@surfacemesh/surfacemesh.m @@ -6,6 +6,7 @@ x y z + ptype end @@ -35,11 +36,15 @@ edgenormals connectivity + basis + coeffs2vals + vals2coeffs + end methods - function dom = surfacemesh(x, y, z) + function dom = surfacemesh(x, y, z, ptype) if ( nargin == 0 ) return @@ -50,9 +55,17 @@ error('X, Y, and Z must be cell arrays of the same size.'); end + if ( nargin < 4 ) + ptype = surfacemesh.patchtype.quad; + end + dom.x = x; dom.y = y; dom.z = z; + dom.ptype = surfacemesh.patchtype(ptype); + if ( length(dom.ptype) == 1 ) %#ok + dom.ptype = repmat(dom.ptype, length(x), 1); + end xu = cell(size(x)); xv = cell(size(x)); yu = cell(size(x)); yv = cell(size(x)); @@ -64,19 +77,46 @@ F = cell(size(x)); G = cell(size(x)); J = cell(size(x)); - - n = size(x{1}, 1); - D = diffmat(n); singular = false(size(x)); + + idx = find(dom.ptype == 'tri'); + if ( any(idx) ) + npts = length(x{idx(1)}); + n = (sqrt(8*npts+1) - 1) / 2; + [xx, yy] = trianglepts(n); + [K, Du, Dv] = koornwinder(n-1, xx, yy); + Du = Du / K; + Dv = Dv / K; + for k = idx(:).' + xu{k} = Du * x{k}; xv{k} = Dv * x{k}; + yu{k} = Du * y{k}; yv{k} = Dv * y{k}; + zu{k} = Du * z{k}; zv{k} = Dv * z{k}; + end + V = inv(K); + dom.coeffs2vals = @(cfs) K*cfs; + dom.vals2coeffs = @(vals) V*vals; %#ok + dom.basis = @(x,y) koornwinder(n-1, x, y); + end + + idx = find(dom.ptype == 'quad'); + if ( any(idx) ) + n = size(x{idx(1)}, 1); + D = diffmat(n); + for k = idx(:).' + xu{k} = x{k} * D.'; xv{k} = D * x{k}; + yu{k} = y{k} * D.'; yv{k} = D * y{k}; + zu{k} = z{k} * D.'; zv{k} = D * z{k}; + end + dom.coeffs2vals = @(cfs) chebtech2.coeffs2vals(chebtech2.coeffs2vals(cfs).').'; + dom.vals2coeffs = @(vals) chebtech2.vals2coeffs(chebtech2.vals2coeffs(vals).').'; + dom.basis = @(x,y) koornwinder(n-1, x, y); + end + for k = 1:length(x) - xu{k} = x{k} * D.'; xv{k} = D * x{k}; - yu{k} = y{k} * D.'; yv{k} = D * y{k}; - zu{k} = z{k} * D.'; zv{k} = D * z{k}; E{k} = xu{k}.*xu{k} + yu{k}.*yu{k} + zu{k}.*zu{k}; G{k} = xv{k}.*xv{k} + yv{k}.*yv{k} + zv{k}.*zv{k}; F{k} = xu{k}.*xv{k} + yu{k}.*yv{k} + zu{k}.*zv{k}; J{k} = E{k}.*G{k} - F{k}.^2; - scl = max(abs(G{k}.*xu{k}-F{k}.*xv{k}), [], 'all'); if ( any(abs(J{k}) < 1e-10*scl, 'all') ) singular(k) = true; @@ -123,6 +163,7 @@ methods ( Static ) dom = sphere(varargin); + dom = icosphere(varargin); dom = ellipsoid(varargin); dom = hemisphere(varargin); dom = blob(varargin); diff --git a/@surfacemesh/volume.m b/@surfacemesh/volume.m index cc3203a..1b4f3c8 100644 --- a/@surfacemesh/volume.m +++ b/@surfacemesh/volume.m @@ -8,16 +8,14 @@ return end -[nv, nu] = size(dom.x{1}); -wu = chebtech2.quadwts(nu); wu = wu(:); -wv = chebtech2.quadwts(nv); wv = wv(:); +W = quadwts(dom); V = 0; for k = 1:length(dom) I = (dom.x{k} .* dom.facenormals{k}(:,:,1) + ... dom.y{k} .* dom.facenormals{k}(:,:,2) + ... dom.z{k} .* dom.facenormals{k}(:,:,3)) / 3; - V = V + sum(sum(I .* wv .* wu.' .* sqrt(dom.J{k}))); + V = V + sum(I .* W(:,:,k), 'all'); end %%% Alternative way: diff --git a/@surfacemesh/wireframe.m b/@surfacemesh/wireframe.m index 22529d4..a61fda4 100644 --- a/@surfacemesh/wireframe.m +++ b/@surfacemesh/wireframe.m @@ -4,46 +4,102 @@ function wireframe(dom, varargin) parser = inputParser; parser.KeepUnmatched = true; parser.addParameter('surface', 'auto', @(s) contains(lower(s), {'auto', 'on', 'off'})); +parser.addParameter('edges', 'auto', @(s) contains(lower(s), {'auto', 'on', 'off'})); parse(parser, varargin{:}); showSurface = parser.Results.surface; +showEdges = parser.Results.edges; varargin = namedargs2cell(parser.Unmatched); -defaultStyle = {'Color', 'k', 'LineStyle', '-', 'LineWidth', 1}; +defaultStyle = {'Color', 'k', ... + 'LineStyle', '-', ... + 'LineWidth', 1}; + +surfaceStyle = {'FaceColor', 'w', ... + 'EdgeColor', 'none', ... + 'AmbientStrength', 0.6, ... + 'DiffuseStrength', 0.4, ... + 'SpecularStrength', 0.3 }; holdState = ishold(); -x = dom.x; -y = dom.y; -z = dom.z; vn = dom.facenormals; +ne = length(dom); +scl = 0.005; + +if ( all(dom.ptype == 'tri') ) + + npts = length(dom.x{1}); + n = (sqrt(8*npts+1)-1) / 2; + x = [dom.x{:}]; + y = [dom.y{:}]; + z = [dom.z{:}]; + e1 = 1:n; + e2 = cumsum([1 (n:-1:2)]); + e3 = cumsum([n (n-1:-1:1)]); + X = [ x(e1,:) ; nan(1,ne) ; x(e2,:) ; nan(1,ne) ; x(e3,:) ; nan(1,ne) ]; + Y = [ y(e1,:) ; nan(1,ne) ; y(e2,:) ; nan(1,ne) ; y(e3,:) ; nan(1,ne) ]; + Z = [ z(e1,:) ; nan(1,ne) ; z(e2,:) ; nan(1,ne) ; z(e3,:) ; nan(1,ne) ]; + + if ( ~strcmpi(showEdges, 'off') ) + plot3(X(:), Y(:), Z(:), defaultStyle{:}, varargin{:}) + end + + % If the plot is not being added to another then plot the surface so + % that the lines are more easily discernable. + if ( (~holdState && strcmpi(showSurface, 'auto')) || strcmpi(showSurface, 'on') ) + % Plot the surface, making it slightly smaller so lines show up + % more clearly. + + T = trilattice(n); + ntri = size(T, 1); + T_all = repmat(T, [1 1 ne]) + reshape((0:ne-1)*npts, 1, 1, []); + T_all = permute(T_all, [1 3 2]); + T_all = reshape(T_all, ntri*ne, 3); + vn_all = reshape([vn{:}], npts*ne, 3); + x_all = x(:) - scl*vn_all(:,1); + y_all = y(:) - scl*vn_all(:,2); + z_all = z(:) - scl*vn_all(:,3); -n = size(x{1}, 1); -ne = length(x); -x = cat(3, x{:}); -y = cat(3, y{:}); -z = cat(3, z{:}); -X = [ x(:,1,:) ; nan(1,1,ne) ; x(:,n,:) ; nan(1,1,ne) ; permute(x(1,:,:), [2 1 3]) ; nan(1,1,ne) ; permute(x(n,:,:), [2 1 3]) ; nan(1,1,ne) ]; -Y = [ y(:,1,:) ; nan(1,1,ne) ; y(:,n,:) ; nan(1,1,ne) ; permute(y(1,:,:), [2 1 3]) ; nan(1,1,ne) ; permute(y(n,:,:), [2 1 3]) ; nan(1,1,ne) ]; -Z = [ z(:,1,:) ; nan(1,1,ne) ; z(:,n,:) ; nan(1,1,ne) ; permute(z(1,:,:), [2 1 3]) ; nan(1,1,ne) ; permute(z(n,:,:), [2 1 3]) ; nan(1,1,ne) ]; -plot3(X(:), Y(:), Z(:), defaultStyle{:}, varargin{:}) - -% If the plot is not being added to another then plot the surface so -% that the lines are more easily discernable. -if ( (~holdState && strcmpi(showSurface, 'auto')) || strcmpi(showSurface, 'on') ) - % Plot the surface, making it slightly smaller so lines show up - % more clearly. - hold on - for k = 1:length(dom) - scl = 0.005; - surface(dom.x{k} - scl*vn{k}(:,:,1), ... - dom.y{k} - scl*vn{k}(:,:,2), ... - dom.z{k} - scl*vn{k}(:,:,3), ... - 0*dom.x{k}, 'FaceColor', 'w', 'EdgeColor', 'none', ... - 'AmbientStrength', 0.6, 'DiffuseStrength', 0.4, 'SpecularStrength', 0.3); + hold on + patch('Faces', T_all, ... + 'Vertices', [x_all y_all z_all], ... + 'FaceVertexCData', 0*x_all, ... + surfaceStyle{:}); end + +elseif ( all(dom.ptype == 'quad') ) + + n = size(dom.x{1}, 1); + x = cat(3, dom.x{:}); + y = cat(3, dom.y{:}); + z = cat(3, dom.z{:}); + X = [ x(:,1,:) ; nan(1,1,ne) ; x(:,n,:) ; nan(1,1,ne) ; permute(x(1,:,:), [2 1 3]) ; nan(1,1,ne) ; permute(x(n,:,:), [2 1 3]) ; nan(1,1,ne) ]; + Y = [ y(:,1,:) ; nan(1,1,ne) ; y(:,n,:) ; nan(1,1,ne) ; permute(y(1,:,:), [2 1 3]) ; nan(1,1,ne) ; permute(y(n,:,:), [2 1 3]) ; nan(1,1,ne) ]; + Z = [ z(:,1,:) ; nan(1,1,ne) ; z(:,n,:) ; nan(1,1,ne) ; permute(z(1,:,:), [2 1 3]) ; nan(1,1,ne) ; permute(z(n,:,:), [2 1 3]) ; nan(1,1,ne) ]; + + if ( ~strcmpi(showEdges, 'off') ) + plot3(X(:), Y(:), Z(:), defaultStyle{:}, varargin{:}) + end + + % If the plot is not being added to another then plot the surface so + % that the lines are more easily discernable. + if ( (~holdState && strcmpi(showSurface, 'auto')) || strcmpi(showSurface, 'on') ) + % Plot the surface, making it slightly smaller so lines show up + % more clearly. + hold on + for k = 1:ne + surface(dom.x{k} - scl*vn{k}(:,:,1), ... + dom.y{k} - scl*vn{k}(:,:,2), ... + dom.z{k} - scl*vn{k}(:,:,3), ... + 0*dom.x{k}, ... + surfaceStyle{:}); + end + end + end if ( ~holdState ) + view(3) axis equal grid on end From 3c5111b0e96470486a4dd3d608c27f6dc1f0f51b Mon Sep 17 00:00:00 2001 From: Dan Fortunato Date: Tue, 28 Oct 2025 18:05:48 -0400 Subject: [PATCH 3/7] Support triangle elements in surfacefun --- @surfacefun/contour.m | 62 +++++++++++++----- @surfacefun/diff.m | 39 +++++++++--- @surfacefun/get.m | 1 + @surfacefun/integral2.m | 6 +- @surfacefun/mesh.m | 57 ++++++++++++++--- @surfacefun/norm.m | 45 +++++++------ @surfacefun/order.m | 6 ++ @surfacefun/private/koornvals2duffyvals.m | 28 +++++++++ @surfacefun/private/koornvals2plotvals.m | 24 +++++++ @surfacefun/resample.m | 28 +++++++-- @surfacefun/subsasgn.m | 8 ++- @surfacefun/surf.m | 77 ++++++++++++++++++----- @surfacefun/surfacefun.m | 4 +- @surfacefun/vec.m | 5 +- 14 files changed, 306 insertions(+), 84 deletions(-) create mode 100644 @surfacefun/order.m create mode 100644 @surfacefun/private/koornvals2duffyvals.m create mode 100644 @surfacefun/private/koornvals2plotvals.m diff --git a/@surfacefun/contour.m b/@surfacefun/contour.m index 94fd5dc..6a085f2 100644 --- a/@surfacefun/contour.m +++ b/@surfacefun/contour.m @@ -35,11 +35,39 @@ function contour(f, varargin) levels = linspace(minu, maxu, N); end +ff = f.vals; +xx = f.domain.x; +yy = f.domain.y; +zz = f.domain.z; +n = order(f)+1; +[bk, ~, vk] = chebpts(n); + +if ( ~all(f.domain.ptype == f.domain.ptype(1)) ) + error('Heterogeneous patch types are not yet supported.'); +end + +if ( f.domain.ptype(1) == 'tri' ) %#ok + % Convert triangle points to tensor-product points using the Duffy + % transformation so that we can use MATLAB's built-in contour(). + [eta1, eta2] = chebpts2(n, n, [0 1 0 1]); + xd = eta1.*(1-eta2); + yd = eta2; + K = koornwinder(n-1); + Kd = koornwinder(n-1, xd, yd); + B = Kd / K; + for k = 1:length(f) + ff{k} = reshape(B*ff{k}, n, n); + xx{k} = reshape(B*xx{k}, n, n); + yy{k} = reshape(B*yy{k}, n, n); + zz{k} = reshape(B*zz{k}, n, n); + end +end + % Loop over the patches: m = 100; [uu, vv] = meshgrid(linspace(-1, 1, m)); for j = 1:length(f) - u = chebvals2plotvals(f.vals{j}); + u = chebvals2plotvals(ff{j}); if ( ~isreal(u) ) u = abs(u); end @@ -59,25 +87,25 @@ function contour(f, varargin) % If the plot is not being added to another then plot the surface so % that the lines are more easily discernable. - if ( ~holdState ) + %if ( ~holdState ) % Plot the surface, making it slightly smaller so lines show up % more clearly. - xx = f.domain.x{j}; - yy = f.domain.y{j}; - zz = f.domain.z{j}; - scl = 0.99; - surf(scl*xx, scl*yy, scl*zz, 1+0*xx, 'FaceColor', 'w', 'EdgeColor', 'None'); - hold on - end + %xx = f.domain.x{j}; + %yy = f.domain.y{j}; + %zz = f.domain.z{j}; + %scl = 0.99; + %surf(scl*xx{j}, scl*yy{j}, scl*zz{j}, 1+0*xx{j}, 'FaceColor', 'w', 'EdgeColor', 'None'); + %hold on + %end % Plot the contours on the surface. k = 1; while ( k < size(C, 2) ) kl = C(2, k); v = k+1:k+kl; - xv = bary2d(f.domain.x{j}, C(1, v), C(2, v)); - yv = bary2d(f.domain.y{j}, C(1, v), C(2, v)); - zv = bary2d(f.domain.z{j}, C(1, v), C(2, v)); + xv = bary2d(xx{j}, C(1, v), C(2, v), bk, vk); + yv = bary2d(yy{j}, C(1, v), C(2, v), bk, vk); + zv = bary2d(zz{j}, C(1, v), C(2, v), bk, vk); % If the line color is a float then we are plotting all contours in % a single color. @@ -93,6 +121,10 @@ function contour(f, varargin) hold on end end + +% Plot the surface, making it slightly smaller so lines show up +plot(f.domain, edges='off', surface='on') + axis equal if ( ~holdState ) @@ -101,12 +133,12 @@ function contour(f, varargin) end -function out = bary2d(vals, x, y) +function out = bary2d(vals, x, y, bk, vk) out = zeros(size(x)); -yvals = bary(y(:), vals).'; +yvals = bary(y(:), vals, bk, vk).'; for k = 1:numel(x) - out(k) = bary(x(k), yvals(:,k)); + out(k) = bary(x(k), yvals(:,k), bk, vk); end out = reshape(out, size(x)); diff --git a/@surfacefun/diff.m b/@surfacefun/diff.m index 60bba94..cf9e494 100644 --- a/@surfacefun/diff.m +++ b/@surfacefun/diff.m @@ -47,28 +47,47 @@ ny = n(2); nz = n(3); -[nv, nu] = size(f(1).vals{1}); -Du = diffmat(nu); -Dv = diffmat(nv); +dom = f(1).domain; +if ( ~all(dom.ptype == dom.ptype(1)) ) + error('Heterogeneous patch types are not yet supported.'); +end + +switch ( dom.ptype(1) ) + case 'tri' + [K, Ku, Kv] = koornwinder(order(dom)); + Du = Ku / K; + Dv = Kv / K; + du = @(x) Du * x; + dv = @(x) Dv * x; + case 'quad' + %[nv, nu] = size(f(1).vals{1}); + [nu, nv] = size(dom); + Du = diffmat(nu); + Dv = diffmat(nv); + du = @(x) x * Du.'; + dv = @(x) Dv * x; +end % TODO: This can be vectorized across multiple functions. nf = builtin('numel', f); for j = 1:nf for k = 1:length(f(j)) vals = f(j).vals{k}; - for m = 1:nx, vals = mappedVDiff(vals, f(j).domain, k, 1, Du, Dv); end - for m = 1:ny, vals = mappedVDiff(vals, f(j).domain, k, 2, Du, Dv); end - for m = 1:nz, vals = mappedVDiff(vals, f(j).domain, k, 3, Du, Dv); end + for m = 1:nx, vals = mappedVDiff(vals, f(j).domain, k, 1, du, dv); end + for m = 1:ny, vals = mappedVDiff(vals, f(j).domain, k, 2, du, dv); end + for m = 1:nz, vals = mappedVDiff(vals, f(j).domain, k, 3, du, dv); end f(j).vals{k} = vals; end end end -function f = mappedVDiff(f, dom, k, dim, Du, Dv) +function f = mappedVDiff(f, dom, k, dim, du, dv) - dfdu = f * Du.'; - dfdv = Dv * f; + dfdu = du(f); + dfdv = dv(f); + %dfdu = f * Du.'; + %dfdv = Dv * f; % Get Jacobian factors for the specified dimension if ( dim == 1 ) @@ -81,7 +100,7 @@ du = dom.uz{k}; dv = dom.vz{k}; else - error('SURFACEFUN:mappedDiff:dim', 'Dimension should be either 1 or 2.'); + error('SURFACEFUN:mappedVDiff:dim', 'Dimension should be either 1 or 2.'); end % Compute df/dx = (du/dx) df/du + (dv/dx) df/dv diff --git a/@surfacefun/get.m b/@surfacefun/get.m index 5ec8629..9f416c7 100644 --- a/@surfacefun/get.m +++ b/@surfacefun/get.m @@ -5,6 +5,7 @@ % % 'DOMAIN' % 'VALS' +% 'VEC' % % See also SUBSREF. diff --git a/@surfacefun/integral2.m b/@surfacefun/integral2.m index 2fe9395..59d7dc0 100644 --- a/@surfacefun/integral2.m +++ b/@surfacefun/integral2.m @@ -24,12 +24,10 @@ % If a patch uses an N x N discretization, then quadrature is performed on % that patch using N points. +W = quadwts(f.domain); I = zeros(length(f), 1); for k = 1:length(f) - [nv, nu] = size(f.vals{k}); - wu = chebtech2.quadwts(nu); wu = wu(:); - wv = chebtech2.quadwts(nv); wv = wv(:); - I(k) = sum(sum(f.vals{k} .* wv .* wu.' .* sqrt(f.domain.J{k}))); + I(k) = sum(f.vals{k} .* W(:,:,k), 'all'); end % Combine norms on each element. diff --git a/@surfacefun/mesh.m b/@surfacefun/mesh.m index 9eb59d5..0aff991 100644 --- a/@surfacefun/mesh.m +++ b/@surfacefun/mesh.m @@ -5,19 +5,60 @@ % % See also PLOT, SURF, CONTOUR. +surfaceStyle = {'FaceColor', 'w', ... + 'AmbientStrength', 0.6, ... + 'DiffuseStrength', 0.4, ... + 'SpecularStrength', 0.3 }; + holdState = ishold(); -[x, y, z] = getGrid(f); -for k = 1:length(f) - u = f.vals{k}; - if ( ~isreal(u) ) - u = abs(u); - end - h(k) = mesh(x{k}, y{k}, z{k}, u, varargin{:}); %#ok - hold on +if ( ~all(f.domain.ptype == f.domain.ptype(1)) ) + error('Heterogeneous patch types are not yet supported.'); +end + +switch ( f.domain.ptype(1) ) + case 'tri' + x = f.domain.x; + y = f.domain.y; + z = f.domain.z; + ne = length(f.domain); + n = order(f.domain)+1; + npts = length(f.domain.x{1}); + T = trilattice(n); + ntri = size(T, 1); + T_all = repmat(T, [1 1 ne]) + reshape((0:ne-1)*npts, 1, 1, []); + T_all = permute(T_all, [1 3 2]); + T_all = reshape(T_all, ntri*ne, 3); + x = [x{:}]; + y = [y{:}]; + z = [z{:}]; + x_all = x(:); + y_all = y(:); + z_all = z(:); + f_all = reshape([f.vals{:}], [], 1); + + hold on + patch('Faces', T_all, ... + 'Vertices', [x_all y_all z_all], ... + 'FaceVertexCData', f_all, ... + 'EdgeColor', 'interp', ... + surfaceStyle{:}, ... + varargin{:}); + + case 'quad' + [x, y, z] = getGrid(f); + for k = 1:length(f) + u = f.vals{k}; + if ( ~isreal(u) ) + u = abs(u); + end + h(k) = mesh(x{k}, y{k}, z{k}, u, varargin{:}); %#ok + hold on + end end if ( ~holdState ) + view(3) axis equal hold off end diff --git a/@surfacefun/norm.m b/@surfacefun/norm.m index ea5f91d..b7703a2 100644 --- a/@surfacefun/norm.m +++ b/@surfacefun/norm.m @@ -111,26 +111,31 @@ % P should be an integer. p = round(p); -% If a patch uses an N x N discretization, then quadrature is performed on -% that patch using N*P points. -int = zeros(length(f), 1); -for k = 1:length(f) - [nv, nu] = size(f.vals{k}); - qu = nu*p; - qv = nv*p; - nv = min(nv, qv); - nu = min(nu, qu); - coeffs = chebtech2.vals2coeffs(chebtech2.vals2coeffs(f.vals{k}).').'; - U = zeros(qv, qu); - U(1:nv,1:nu) = coeffs(1:nv,1:nu); - V = chebtech2.coeffs2vals(chebtech2.coeffs2vals(U).').'; - J_cfs = chebtech2.vals2coeffs(chebtech2.vals2coeffs(f.domain.J{k}).').'; - J_q = zeros(qv, qu); - J_q(1:nv,1:nu) = J_cfs(1:nv,1:nu); - J = chebtech2.coeffs2vals(chebtech2.coeffs2vals(J_q).').'; - wu = chebtech2.quadwts(qu); wu = wu(:); - wv = chebtech2.quadwts(qv); wv = wv(:); - int(k) = sum(sum(abs(V).^p .* wv .* wu.' .* sqrt(J))).^(1/p); +if ( any(f.domain.ptype == 'tri') ) + int = integral(abs(f).^p, 'all').^(1/p); +else + % Use upsampled quadrature for dealiasing on a quadrilateral patch. + % If a patch uses an N x N discretization, then quadrature is performed on + % that patch using N*P points. + int = zeros(length(f), 1); + for k = 1:length(f) + [nv, nu] = size(f.vals{k}); + qu = nu*p; + qv = nv*p; + nv = min(nv, qv); + nu = min(nu, qu); + coeffs = chebtech2.vals2coeffs(chebtech2.vals2coeffs(f.vals{k}).').'; + U = zeros(qv, qu); + U(1:nv,1:nu) = coeffs(1:nv,1:nu); + V = chebtech2.coeffs2vals(chebtech2.coeffs2vals(U).').'; + J_cfs = chebtech2.vals2coeffs(chebtech2.vals2coeffs(f.domain.J{k}).').'; + J_q = zeros(qv, qu); + J_q(1:nv,1:nu) = J_cfs(1:nv,1:nu); + J = chebtech2.coeffs2vals(chebtech2.coeffs2vals(J_q).').'; + wu = chebtech2.quadwts(qu); wu = wu(:); + wv = chebtech2.quadwts(qv); wv = wv(:); + int(k) = sum(sum(abs(V).^p .* wv .* wu.' .* sqrt(J))).^(1/p); + end end end diff --git a/@surfacefun/order.m b/@surfacefun/order.m new file mode 100644 index 0000000..0a495ca --- /dev/null +++ b/@surfacefun/order.m @@ -0,0 +1,6 @@ +function [pu, pv] = order(f) +%ORDER Order of a SURFACEFUN. + +[pu, pv] = order(f.domain); + +end diff --git a/@surfacefun/private/koornvals2duffyvals.m b/@surfacefun/private/koornvals2duffyvals.m new file mode 100644 index 0000000..5dd705b --- /dev/null +++ b/@surfacefun/private/koornvals2duffyvals.m @@ -0,0 +1,28 @@ +function vals = koornvals2duffyvals(vals, nplotpts) +%KOORNVALS2PLOTVALS Convert Koornwinder coefficients to values at equispaced +% tensor-product points. + +persistent Eval nstored nplotptsstored + +if ( nargin < 2 ) + nplotpts = 100; +end + +npts = length(vals); +n = (sqrt(8*npts+1)-1) / 2; +if ( isempty(Eval) || n ~= nstored || nplotpts ~= nplotptsstored ) + nstored = n; + nplotptsstored = nplotpts; + [x, y] = trianglepts(n); + [eta1, eta2] = meshgrid(linspace(0, 1, nplotpts)); + % Duffy transformation + xplot = eta1.*(1-eta2); + yplot = eta2; + K = koornwinder(n-1, x, y); + Kplot = koornwinder(n-1, xplot, yplot); + Eval = Kplot / K; +end + +vals = Eval * vals; + +end diff --git a/@surfacefun/private/koornvals2plotvals.m b/@surfacefun/private/koornvals2plotvals.m new file mode 100644 index 0000000..0cd2ebd --- /dev/null +++ b/@surfacefun/private/koornvals2plotvals.m @@ -0,0 +1,24 @@ +function vals = koornvals2plotvals(vals, nplotpts) +%KOORNVALS2PLOTVALS Convert Koornwinder coefficients to values at equispaced points. + +persistent Eval nstored nplotptsstored + +if ( nargin < 2 ) + nplotpts = 100; +end + +npts = length(vals); +n = (sqrt(8*npts+1)-1) / 2; +if ( isempty(Eval) || n ~= nstored || nplotpts ~= nplotptsstored ) + nstored = n; + nplotptsstored = nplotpts; + [x, y] = trianglepts(n); + [xplot, yplot] = trianglepts(nplotpts, type='linspace'); + K = koornwinder(n-1, x, y); + Kplot = koornwinder(n-1, xplot, yplot); + Eval = Kplot / K; +end + +vals = Eval * vals; + +end diff --git a/@surfacefun/resample.m b/@surfacefun/resample.m index 0417c19..3aab8a6 100644 --- a/@surfacefun/resample.m +++ b/@surfacefun/resample.m @@ -4,12 +4,30 @@ return end -m = size(u.vals{1}, 1); -B = barymat(chebpts(n), chebpts(m)); +if ( ~all(u.domain.ptype == u.domain.ptype(1)) ) + error('Heterogeneous patch types are not yet supported.'); +end + +switch ( u.domain.ptype(1) ) + case 'tri' + m = order(u)+1; + [xm, ym] = trianglepts(m); + [xn, yn] = trianglepts(n); + Km = koornwinder(m-1, xm, ym); + Knm = koornwinder(m-1, xn, yn); + B = Knm / Km; + vals = u.vals; + for k = 1:length(u) + vals{k} = B * vals{k}; + end -vals = u.vals; -for k = 1:length(u) - vals{k} = B * vals{k} * B.'; + case 'quad' + m = size(u.vals{1}, 1); + B = barymat(chebpts(n), chebpts(m)); + vals = u.vals; + for k = 1:length(u) + vals{k} = B * vals{k} * B.'; + end end dom = resample(u.domain, n); diff --git a/@surfacefun/subsasgn.m b/@surfacefun/subsasgn.m index e1bcac8..5981986 100644 --- a/@surfacefun/subsasgn.m +++ b/@surfacefun/subsasgn.m @@ -11,16 +11,18 @@ switch index(1).type case '.' if ( strcmpi(index(1).subs, 'vec') ) - n = order(f(1).domain) + 1; + [nu, nv] = size(f(1).domain); + npts = nu*nv; numPatches = length(f(1).domain); + numDOF = numel(f(1)); numFuns = size(f, 2); - if ( ~all(size(val) == [n^2*numPatches numFuns]) ) + if ( ~all(size(val) == [numDOF numFuns]) ) error('SURFACEFUN:subsasgn:dimensions', ... 'Data and surfacefun dimensions are incompatible.'); end for j = 1:numFuns for k = 1:numPatches - f(:,j).vals{k}(:) = val((k-1)*n^2+(1:n^2), j); + f(:,j).vals{k}(:) = val((k-1)*npts+(1:npts), j); end end else diff --git a/@surfacefun/surf.m b/@surfacefun/surf.m index 8eba600..5db9fd5 100644 --- a/@surfacefun/surf.m +++ b/@surfacefun/surf.m @@ -33,28 +33,75 @@ function surf(f, varargin) holdState = ishold(); -for k = 1:length(f) - u = f.vals{k}; - if ( ~isreal(u) ) - u = real(u); - end - x = f.domain.x{k}; - y = f.domain.y{k}; - z = f.domain.z{k}; +if ( ~all(f.domain.ptype == f.domain.ptype(1)) ) + error('Heterogeneous patch types are not yet supported.'); +end + +switch ( f.domain.ptype(1) ) + + case 'tri' + + x = f.domain.x; + y = f.domain.y; + z = f.domain.z; + u = f.vals; + + for k = 1:length(f) + x{k} = koornvals2plotvals(x{k}, nplotpts); + y{k} = koornvals2plotvals(y{k}, nplotpts); + z{k} = koornvals2plotvals(z{k}, nplotpts); + u{k} = koornvals2plotvals(u{k}, nplotpts); + end + + x_all = reshape([x{:}], [], 1); + y_all = reshape([y{:}], [], 1); + z_all = reshape([z{:}], [], 1); + u_all = reshape([u{:}], [], 1); + + if ( ~isreal(u_all) ) + u_all = real(u_all); + end + + T = trilattice(nplotpts); + ntri = size(T, 1); + ne = length(u); + npts = length(u{1}); + T_all = repmat(T, [1 1 ne]) + reshape((0:ne-1)*npts, 1, 1, []); + T_all = permute(T_all, [1 3 2]); + T_all = reshape(T_all, ntri*ne, 3); + + hold on + patch('Faces', T_all, ... + 'Vertices', [x_all y_all z_all], ... + 'FaceVertexCData', u_all, ... + varargin{:}); - u = chebvals2plotvals(u, nplotpts); - x = chebvals2plotvals(x, nplotpts); - y = chebvals2plotvals(y, nplotpts); - z = chebvals2plotvals(z, nplotpts); + case 'quad' - surface(x, y, z, u, varargin{:}); - hold on + for k = 1:length(f) + u = f.vals{k}; + if ( ~isreal(u) ) + u = real(u); + end + x = f.domain.x{k}; + y = f.domain.y{k}; + z = f.domain.z{k}; + + u = chebvals2plotvals(u, nplotpts); + x = chebvals2plotvals(x, nplotpts); + y = chebvals2plotvals(y, nplotpts); + z = chebvals2plotvals(z, nplotpts); + + surface(x, y, z, u, varargin{:}); + hold on + end end if ( ~holdState ) view(3) shading interp - axis equal + %axis equal + set(gca, 'DataAspectRatio', [1 1 1], 'PlotBoxAspectRatioMode', 'auto') grid on hold off end diff --git a/@surfacefun/surfacefun.m b/@surfacefun/surfacefun.m index 44a1e9f..a7aa083 100644 --- a/@surfacefun/surfacefun.m +++ b/@surfacefun/surfacefun.m @@ -68,12 +68,12 @@ obj.domain = dom; if ( isempty(vals) ) - n = order(dom)+1; + [nu, nv] = size(dom); xx = reshape([dom.x{:}], [], 1); yy = reshape([dom.y{:}], [], 1); zz = reshape([dom.z{:}], [], 1); ff = feval(func, xx, yy, zz); - ff = reshape(ff, [n n length(dom)]); + ff = reshape(ff, [nv nu length(dom)]); vals = cell(length(dom), 1); for k = 1:length(dom) vals{k} = ff(:,:,k); diff --git a/@surfacefun/vec.m b/@surfacefun/vec.m index e3ebe37..5bc0c87 100644 --- a/@surfacefun/vec.m +++ b/@surfacefun/vec.m @@ -7,7 +7,8 @@ % % See also SUBSASGN. -n = order(f(1).domain) + 1; +[nu, nv] = size(f(1).domain); +npts = nu*nv; numPatches = length(f(1).domain); numDOF = numel(f(1)); numFuns = size(f, 2); @@ -15,7 +16,7 @@ x = zeros(numDOF, numFuns); for j = 1:numFuns for k = 1:numPatches - x((k-1)*n^2+(1:n^2), j) = f(:,j).vals{k}(:); + x((k-1)*npts+(1:npts), j) = f(:,j).vals{k}(:); end end From 6365fc3d7e5b1fb049f39e25651f4fca58ee834f Mon Sep 17 00:00:00 2001 From: Dan Fortunato Date: Tue, 28 Oct 2025 18:09:17 -0400 Subject: [PATCH 4/7] Add quiver plotting for triangle meshes --- @surfacefunv/quiver.m | 76 +++++++++++++++++++++++++++++++------------ 1 file changed, 56 insertions(+), 20 deletions(-) diff --git a/@surfacefunv/quiver.m b/@surfacefunv/quiver.m index 17f4fa4..b6adeb9 100644 --- a/@surfacefunv/quiver.m +++ b/@surfacefunv/quiver.m @@ -33,26 +33,62 @@ z = dom.z; vn = dom.facenormals; -xx = zeros(m, m, length(dom)); fx = zeros(m, m, length(dom)); -yy = zeros(m, m, length(dom)); fy = zeros(m, m, length(dom)); -zz = zeros(m, m, length(dom)); fz = zeros(m, m, length(dom)); -n = size(x{1}, 1); -B = bary(linspace(-1, 1, m).', eye(n)); -for k = 1:length(dom) - xx(:,:,k) = B * x{k} * B.'; - yy(:,:,k) = B * y{k} * B.'; - zz(:,:,k) = B * z{k} * B.'; - fx(:,:,k) = B * f.components{1}.vals{k} * B.'; - fy(:,:,k) = B * f.components{2}.vals{k} * B.'; - fz(:,:,k) = B * f.components{3}.vals{k} * B.'; - if ( ~holdState ) - r = 0.01; - surface(x{k} - r*vn{k}(:,:,1), ... - y{k} - r*vn{k}(:,:,2), ... - z{k} - r*vn{k}(:,:,3), ... - 0*x{k}, 'FaceColor', 'w', 'EdgeColor', 'None'); - hold on - end +if ( ~all(f.domain.ptype == f.domain.ptype(1)) ) + error('Heterogeneous patch types are not yet supported.'); +end + +switch ( dom.ptype(1) ) + case 'tri' + npts = m*(m+1)/2; + n = (sqrt(8*length(x{1})+1)-1)/2; + [uu, vv] = trianglepts(n, type='cheb'); + [uuu, vvv] = trianglepts(m, type='equi'); + Vcheb = koornwinder(n-1, uu, vv); + Vequi = koornwinder(n-1, uuu, vvv); + cheb2equi = Vequi / Vcheb; + + xx = zeros(npts, length(dom)); fx = zeros(npts, length(dom)); + yy = zeros(npts, length(dom)); fy = zeros(npts, length(dom)); + zz = zeros(npts, length(dom)); fz = zeros(npts, length(dom)); + for k = 1:length(dom) + xx(:,k) = cheb2equi * x{k}; + yy(:,k) = cheb2equi * y{k}; + zz(:,k) = cheb2equi * z{k}; + fx(:,k) = cheb2equi * f.components{1}.vals{k}; + fy(:,k) = cheb2equi * f.components{2}.vals{k}; + fz(:,k) = cheb2equi * f.components{3}.vals{k}; + end + + if ( ~holdState ) + hold on + plot(dom, edges='off', surface='on') + end + + case 'quad' + xx = zeros(m, m, length(dom)); fx = zeros(m, m, length(dom)); + yy = zeros(m, m, length(dom)); fy = zeros(m, m, length(dom)); + zz = zeros(m, m, length(dom)); fz = zeros(m, m, length(dom)); + n = size(x{1}, 1); + B = bary(linspace(-1, 1, m).', eye(n)); + for k = 1:length(dom) + xx(:,:,k) = B * x{k} * B.'; + yy(:,:,k) = B * y{k} * B.'; + zz(:,:,k) = B * z{k} * B.'; + fx(:,:,k) = B * f.components{1}.vals{k} * B.'; + fy(:,:,k) = B * f.components{2}.vals{k} * B.'; + fz(:,:,k) = B * f.components{3}.vals{k} * B.'; + if ( ~holdState ) + r = 0.01; + surface(x{k} - r*vn{k}(:,:,1), ... + y{k} - r*vn{k}(:,:,2), ... + z{k} - r*vn{k}(:,:,3), ... + 0*x{k}, 'FaceColor', 'w', 'EdgeColor', 'None'); + hold on + end + end + + otherwise + error('Unknown patch type.'); end h = quiver3(xx, yy, zz, scl*fx, scl*fy, scl*fz, 0, varargin{:}); From 1df81c584507ac92a4c52086a8086154c522c1eb Mon Sep 17 00:00:00 2001 From: Dan Fortunato Date: Wed, 29 Oct 2025 11:39:48 -0400 Subject: [PATCH 5/7] Support triangles in surfaceop --- ...initialize_DtN.m => initialize_DtN_quad.m} | 22 +- +surfaceop/@leaf/initialize_DtN_tri.m | 327 +++++++++++++++ ...initialize_ItI.m => initialize_ItI_quad.m} | 23 +- +surfaceop/@leaf/initialize_ItI_tri.m | 372 ++++++++++++++++++ +surfaceop/@leaf/leaf.m | 6 +- +surfaceop/@leaf/solve_DtN.m | 4 +- +surfaceop/@leaf/solve_ItI.m | 4 +- +surfaceop/@leaf/updateRHS_DtN.m | 29 +- +surfaceop/@leaf/updateRHS_ItI.m | 25 +- +surfaceop/@parent/solve_DtN.m | 4 +- +surfaceop/@parent/solve_ItI.m | 4 +- +surfaceop/@patch/intersect.m | 18 +- @surfaceop/initialize.m | 14 +- 13 files changed, 791 insertions(+), 61 deletions(-) rename +surfaceop/@leaf/{initialize_DtN.m => initialize_DtN_quad.m} (94%) create mode 100644 +surfaceop/@leaf/initialize_DtN_tri.m rename +surfaceop/@leaf/{initialize_ItI.m => initialize_ItI_quad.m} (95%) create mode 100644 +surfaceop/@leaf/initialize_ItI_tri.m diff --git a/+surfaceop/@leaf/initialize_DtN.m b/+surfaceop/@leaf/initialize_DtN_quad.m similarity index 94% rename from +surfaceop/@leaf/initialize_DtN.m rename to +surfaceop/@leaf/initialize_DtN_quad.m index 34091f5..f78ac34 100644 --- a/+surfaceop/@leaf/initialize_DtN.m +++ b/+surfaceop/@leaf/initialize_DtN_quad.m @@ -1,4 +1,4 @@ -function L = initialize_DtN(op, dom, rhs) +function L = initialize_DtN_quad(op, dom, rhs) %INITIALIZE Initialize an array of LEAF objects. % L = SURFACEOP.LEAF.INITIALIZE(OP, DOM) returns a cell array L of LEAF % objects which contain the solution and D2N operators for Poisson's @@ -24,6 +24,7 @@ numPatches = length(dom); n = size(dom.x{1}, 1); +npts = n^2; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %%%%%%%%%%%%%%%%%%%%%%% DEFINE REFERENCE GRID %%%%%%%%%%%%%%%%%%%%%%%%%% @@ -36,7 +37,6 @@ % Skeleton mappings nskel = n-2; -%nskel = n; numSkelPts = 4*nskel; S2L = skel2leaf(n, nskel); % Don't sparsify for speed L2S = leaf2skel(nskel, n); @@ -60,12 +60,12 @@ NU = pagemtimes(B, NU); NN = [NL ; NR ; ND ; NU]; -ux = reshape([dom.ux{:}], [n^2 numPatches]); vx = reshape([dom.vx{:}], [n^2 numPatches]); -uy = reshape([dom.uy{:}], [n^2 numPatches]); vy = reshape([dom.vy{:}], [n^2 numPatches]); -uz = reshape([dom.uz{:}], [n^2 numPatches]); vz = reshape([dom.vz{:}], [n^2 numPatches]); +ux = reshape([dom.ux{:}], [npts numPatches]); vx = reshape([dom.vx{:}], [npts numPatches]); +uy = reshape([dom.uy{:}], [npts numPatches]); vy = reshape([dom.vy{:}], [npts numPatches]); +uz = reshape([dom.uz{:}], [npts numPatches]); vz = reshape([dom.vz{:}], [npts numPatches]); nrhs = size(rhs, 2); -tmpS = zeros(n^2, numBdyPts+nrhs); +tmpS = zeros(npts, numBdyPts+nrhs); tmpS(ee,:) = eye(numBdyPts, numBdyPts+nrhs); D2N_scl0 = {ones(nskel,1) ; ones(nskel,1) ; ones(nskel,1) ; ones(nskel,1)}; @@ -78,9 +78,9 @@ Du = kron(D, I); Dv = kron(I, D); -X = reshape([dom.x{:}], [n^2 numPatches]); -Y = reshape([dom.y{:}], [n^2 numPatches]); -Z = reshape([dom.z{:}], [n^2 numPatches]); +X = reshape([dom.x{:}], [npts numPatches]); +Y = reshape([dom.y{:}], [npts numPatches]); +Z = reshape([dom.z{:}], [npts numPatches]); flags = structfun(@(f) ~(isscalar(f) && isnumeric(f) && f==0), op, 'UniformOutput', false); @@ -103,7 +103,7 @@ rhs = feval(rhs, X(ii,:), Y(ii,:), Z(ii,:)); rhs = reshape(rhs, [numIntPts 1 numPatches]); elseif ( isa(rhs, 'surfacefun') ) - rhs = reshape(rhs.vec(), [n^2 numPatches nrhs]); + rhs = reshape(rhs.vec(), [npts numPatches nrhs]); rhs = permute(rhs, [1 3 2]); rhs = rhs(ii,:,:); elseif ( isnumeric(rhs) && isscalar(rhs) ) @@ -128,7 +128,7 @@ x(1,1) y(1,1) z(1,1) x(1,n) y(1,n) z(1,n) nskel ; % "Down" side x(n,1) y(n,1) z(n,1) x(n,n) y(n,n) z(n,n) nskel ]; % "Up" side - A = zeros(n^2); + A = zeros(npts); Dx = ux(:,k).*Du + vx(:,k).*Dv; Dy = uy(:,k).*Du + vy(:,k).*Dv; Dz = uz(:,k).*Du + vz(:,k).*Dv; diff --git a/+surfaceop/@leaf/initialize_DtN_tri.m b/+surfaceop/@leaf/initialize_DtN_tri.m new file mode 100644 index 0000000..1477ff6 --- /dev/null +++ b/+surfaceop/@leaf/initialize_DtN_tri.m @@ -0,0 +1,327 @@ +function L = initialize_DtN_tri(op, dom, rhs) +%INITIALIZE Initialize an array of LEAF objects. +% L = SURFACEOP.LEAF.INITIALIZE(OP, DOM) returns a cell array L of LEAF +% objects which contain the solution and D2N operators for Poisson's +% equation on the domain DOM with zero righthand side. +% +% L = SURFACEOP.LEAF.INITIALIZE(OP, DOM, RHS) is as above, but with the +% righthand side RHS, which may be a scalar or a function handle. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %%%%%%%%%%%%%%%%%%%%%%%%%%% PARSE INPUTS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +if ( isempty(dom) ) + L = []; + return +end + +assert(isa(dom, 'surfacemesh'), 'Invalid domain.'); + +if ( nargin < 3 ) + % Default to homogeneous problem: + rhs = 0; +end + +numPatches = length(dom); +npts = length(dom.x{1}); +n = order(dom)+1; + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %%%%%%%%%%%%%%%%%%%%%%% DEFINE REFERENCE GRID %%%%%%%%%%%%%%%%%%%%%%%%%% + +[xx, yy] = trianglepts(n); +eleft = 1:n; +edown = cumsum([1 (n:-1:2)]); +ehypot = cumsum([n (n-1:-1:1)]); +eeIdx = unique([eleft edown ehypot]); +ee = false(size(xx)); +ee(eeIdx) = true; % Boundary indices +ii = ~ee; % Interior indices +numBdyPts = sum(ee(:)); +numIntPts = sum(ii(:)); + +% Skeleton mappings +nskel = n-2; +numSkelPts = 3*nskel; +S2L = skel2leaf(n, nskel); % Don't sparsify for speed +L2S = leaf2skel(nskel, n); +xskel = chebpts(nskel, 1); +[xleaf, ~, wleaf] = chebpts(n, 2); +B = barymat(xskel, xleaf, wleaf); +w = chebtech1.quadwts(nskel); w = w(:); +wskel = [0.5*w ; 0.5*w ; sqrt(2)/2*w]; + +% Skeleton indices for each side +leftSkel = 1:nskel; +downSkel = nskel+1:2*nskel; +hypotSkel = 2*nskel+1:3*nskel; + +% Compute binormal vectors on the skeleton +[NL, ND, NH] = binormals(dom); +NL = pagemtimes(B, NL); +ND = pagemtimes(B, ND); +NH = pagemtimes(B, NH); +NN = [NL ; ND ; NH]; + +ux = reshape([dom.ux{:}], [npts numPatches]); vx = reshape([dom.vx{:}], [npts numPatches]); +uy = reshape([dom.uy{:}], [npts numPatches]); vy = reshape([dom.vy{:}], [npts numPatches]); +uz = reshape([dom.uz{:}], [npts numPatches]); vz = reshape([dom.vz{:}], [npts numPatches]); + +nrhs = size(rhs, 2); +tmpS = zeros(npts, numBdyPts+nrhs); +tmpS(ee,:) = eye(numBdyPts, numBdyPts+nrhs); +D2N_scl0 = {ones(nskel,1) ; ones(nskel,1) ; ones(nskel,1)}; + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %%%%%%%%%%%%%%%%%%%%%%%%% DEFINE OPERATORS %%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +[K, Ku, Kv] = koornwinder(n-1, xx, yy); +Du = Ku / K; +Dv = Kv / K; +II = eye(npts); + +X = reshape([dom.x{:}], [npts numPatches]); +Y = reshape([dom.y{:}], [npts numPatches]); +Z = reshape([dom.z{:}], [npts numPatches]); + +flags = structfun(@(f) ~(isscalar(f) && isnumeric(f) && f==0), op, 'UniformOutput', false); + +for name = fieldnames(op).' + name = name{1}; + if ( isa(op.(name), 'function_handle') ) + op.(name) = feval(op.(name), X, Y, Z); + elseif ( isa(op.(name), 'surfacefun') ) + op.(name) = reshape([op.(name).vals{:}], [npts numPatches]); + elseif ( isscalar(op.(name)) ) + op.(name) = repmat(op.(name), [1 numPatches]); + end +end + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %%%%%%%%%%%%%%%%%%%%%%%%%%% CONSTANT RHS? %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% Evaluate non-constant RHSs if required: +if ( isa(rhs, 'function_handle') ) + rhs = feval(rhs, X(ii,:), Y(ii,:), Z(ii,:)); + rhs = reshape(rhs, [numIntPts 1 numPatches]); +elseif ( isa(rhs, 'surfacefun') ) + rhs = reshape(rhs.vec(), [npts numPatches nrhs]); + rhs = permute(rhs, [1 3 2]); + rhs = rhs(ii,:,:); +elseif ( isnumeric(rhs) && isscalar(rhs) ) + rhs = repmat(rhs, [numIntPts 1 numPatches]); +end + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %%%%%%%%%%%%%%%%%%%%%%% SOLVE LOCAL PROBLEMS %%%%%%%%%%%%%%%%%%%%%%%%%%% + +% Initialize +L = cell(numPatches, 1); + +% Loop over each patch: +for k = 1:numPatches + + % Define the left and right edges for this patch: + x = dom.x{k}; + y = dom.y{k}; + z = dom.z{k}; + edges = [ x(1) y(1) z(1) x(n) y(n) z(n) nskel ; % "Left" side + x(1) y(1) z(1) x(end) y(end) z(end) nskel ; % "Down" side + x(n) y(n) z(n) x(end) y(end) z(end) nskel ]; % "Hypot" side + + A = zeros(npts); + Dx = ux(:,k).*Du + vx(:,k).*Dv; + Dy = uy(:,k).*Du + vy(:,k).*Dv; + Dz = uz(:,k).*Du + vz(:,k).*Dv; + J = dom.J{k}(:); + + if ( dom.singular(k) ) + + % Assemble matrix: + if ( flags.dxx ), A = A + op.dxx(:,k).*(J.*(Dx*Dx)-(Dx*J).*Dx); end + if ( flags.dyy ), A = A + op.dyy(:,k).*(J.*(Dy*Dy)-(Dy*J).*Dy); end + if ( flags.dzz ), A = A + op.dzz(:,k).*(J.*(Dz*Dz)-(Dz*J).*Dz); end + if ( flags.dxy ), A = A + op.dxy(:,k).*(J.*(Dx*Dy)-(Dx*J).*Dy); end + if ( flags.dyx ), A = A + op.dyx(:,k).*(J.*(Dy*Dx)-(Dy*J).*Dx); end + if ( flags.dyz ), A = A + op.dyz(:,k).*(J.*(Dy*Dz)-(Dy*J).*Dz); end + if ( flags.dzy ), A = A + op.dzy(:,k).*(J.*(Dz*Dy)-(Dz*J).*Dy); end + if ( flags.dxz ), A = A + op.dxz(:,k).*(J.*(Dx*Dz)-(Dx*J).*Dz); end + if ( flags.dzx ), A = A + op.dzx(:,k).*(J.*(Dz*Dx)-(Dz*J).*Dx); end + if ( flags.dx ), A = A + op.dx(:,k).*J.^2.*Dx; end + if ( flags.dy ), A = A + op.dy(:,k).*J.^2.*Dy; end + if ( flags.dz ), A = A + op.dz(:,k).*J.^2.*Dz; end + if ( flags.b ), A = A + op.b(:,k).*J.^3.*II; end + + % Construct solution operator: + dA = decomposition(A(ii,ii), 'cod'); + Ainv = @(u) dA \ (J(ii).^3.*u); + S = dA \ ([-A(ii,ee), J(ii).^3.*rhs(:,k)]); + + dx = L2S * (J(ee).^2.*Dx(ee,:)); + dy = L2S * (J(ee).^2.*Dy(ee,:)); + dz = L2S * (J(ee).^2.*Dz(ee,:)); + + % The D2N map needs to be scaled on each side (e.g. when being + % merged) to account for the Jacobian scaling which has been + % factored out of the coordinate derivative maps. This scaling + % is not known until the merge stage, as it depends on the + % scaling of the neighboring patch. + Jss = L2S * J(ee).^3; + D2N_scl = {Jss(leftSkel); Jss(downSkel); Jss(hypotSkel)}; + + else + + % Assemble matrix: + if ( flags.dxx ), A = A + op.dxx(:,k).*(Dx*Dx); end + if ( flags.dyy ), A = A + op.dyy(:,k).*(Dy*Dy); end + if ( flags.dzz ), A = A + op.dzz(:,k).*(Dz*Dz); end + if ( flags.dxy ), A = A + op.dxy(:,k).*(Dx*Dy); end + if ( flags.dyx ), A = A + op.dyx(:,k).*(Dy*Dx); end + if ( flags.dyz ), A = A + op.dyz(:,k).*(Dy*Dz); end + if ( flags.dzy ), A = A + op.dzy(:,k).*(Dz*Dy); end + if ( flags.dxz ), A = A + op.dxz(:,k).*(Dx*Dz); end + if ( flags.dzx ), A = A + op.dzx(:,k).*(Dz*Dx); end + if ( flags.dx ), A = A + op.dx(:,k).*Dx; end + if ( flags.dy ), A = A + op.dy(:,k).*Dy; end + if ( flags.dz ), A = A + op.dz(:,k).*Dz; end + if ( flags.b ), A = A + op.b(:,k).*II; end + + % Construct solution operator: + dA = matlab.internal.decomposition.DenseLU(A(ii,ii)); + Ainv = @(u) solve(dA, u, false); + S = Ainv([-A(ii,ee), rhs(:,:,k)]); + + dx = L2S * Dx(ee,:); + dy = L2S * Dy(ee,:); + dz = L2S * Dz(ee,:); + D2N_scl = D2N_scl0; + end + + % Append boundary points to solution operator and extract the + % particular solution to store separately: + tmpS(ii,:) = S; + S = tmpS(:,1:numBdyPts) * S2L; + u_part = tmpS(:,numBdyPts+1:end); + + % Construct normal derivative operator: + normal_d = NN(:,1,k).*dx + NN(:,2,k).*dy + NN(:,3,k).*dz; + + % Construct the D2N map and particular flux; + D2N = normal_d * S; + du_part = normal_d * u_part; + + JJ = L2S * sqrt(J(ee)); + ww = wskel .* JJ; + xyz = L2S * [x(ee) y(ee) z(ee)]; + + % Assemble the patch: + L{k} = surfaceop.leaf(dom, n, k, S, D2N, D2N_scl, u_part, du_part, edges, xyz, ww, Ainv, normal_d); + +end + +end + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %%%%%%%%%%%%%%%%%%%%%%%%%% DEFINE OPERATORS %%%%%%%%%%%%%%%%%%%%%%%%%%%% + +function [nl, nd, nh] = binormals(dom) +%BINORMALS Compute the binormal vectors for a surfacemesh. + +n = order(dom)+1; +eleft = 1:n; +edown = cumsum([1 (n:-1:2)]); +ehypot = cumsum([n (n-1:-1:1)]); + +npts = size(dom.x{1}, 1); +sz = [npts 1 length(dom)]; +xu = reshape([dom.xu{:}], sz); xv = reshape([dom.xv{:}], sz); +yu = reshape([dom.yu{:}], sz); yv = reshape([dom.yv{:}], sz); +zu = reshape([dom.zu{:}], sz); zv = reshape([dom.zv{:}], sz); + +% Normal vectors to the surface (unnormalized) +nl = -[xu(eleft,:,:) yu(eleft,:,:) zu(eleft,:,:)]; +nd = -[xv(edown,:,:) yv(edown,:,:) zv(edown,:,:)]; +nh = [xu(ehypot,:,:) yu(ehypot,:,:) zu(ehypot,:,:)] + ... + [xv(ehypot,:,:) yv(ehypot,:,:) zv(ehypot,:,:)]; + +% Tangent vectors to the element boundary (normalized) +tl = normalize([xv(eleft,:,:) yv(eleft,:,:) zv(eleft,:,:)]); +td = normalize([xu(edown,:,:) yu(edown,:,:) zu(edown,:,:)]); +th = normalize( -[xv(ehypot,:,:) yv(ehypot,:,:) zv(ehypot,:,:)] + ... + [xu(ehypot,:,:) yu(ehypot,:,:) zu(ehypot,:,:)] ); + +% Binormal vectors (normalized) +nl = normalize(nl - tl .* sum(nl.*tl, 2)); +nd = normalize(nd - td .* sum(nd.*td, 2)); +nh = normalize(nh - th .* sum(nh.*th, 2)); + +end + +function v = normalize(v) + +v = v ./ sqrt(v(:,1,:).^2 + v(:,2,:).^2 + v(:,3,:).^2); + +end + +function P = skel2leaf(nleaf, nskel) +%SKEL2LEAF Boundary interpolation matrix. +% SKEL2LEAF(NLEAF, NSKEL) returns the (3*NLEAF-3) x 3*NSKEL matrix that +% maps 3 pieces of length-NSKEL first-kind boundary values to 3*NLEAF-3 +% second-kind boundary values, including the corners. At each corner, the +% average of the two interpolated values is used. + +[xskel, ~, wskel] = chebpts(nskel, 1); +[xleaf, ~, wleaf] = chebpts(nleaf, 2); +B = barymat(xleaf, xskel, wskel); + +% Skeleton indices for each side +leftSkel = 1:nskel; +downSkel = nskel+1:2*nskel; +hypotSkel = 2*nskel+1:3*nskel; + +% Leaf indices for each side +leftLeaf = 1:nleaf; +%downLeaf = [1 nleaf+1:2*nleaf-1]; +downLeaf = [1 nleaf+1:2:3*nleaf-3]; +%hypotLeaf = [nleaf 2*nleaf:3*nleaf-3 2*nleaf-1]; +hypotLeaf = [nleaf:2:3*nleaf-4 3*nleaf-3]; + +P = zeros(3*nleaf-3, 3*nskel); +P(leftLeaf, leftSkel) = B; +P(downLeaf, downSkel) = B; +P(hypotLeaf, hypotSkel) = B; + +% Average the corners: +corners = [1 nleaf 3*nleaf-3]; +P(corners,:) = P(corners,:)/2; + +end + +function P = leaf2skel(nskel, nleaf) +%LEAF2SKEL Boundary interpolation matrix. +% LEAF2SKEL(NSKEL, NLEAF) returns the 3*NSKEL x (3*NLEAF-3) matrix that +% maps 3*NLEAF-3 second-kind boundary values to 3 pieces of length-NSKEL +% first-kind boundary values. + +[xskel, ~, wskel] = chebpts(nskel, 1); +[xleaf, ~, wleaf] = chebpts(nleaf, 2); +B = barymat(xskel, xleaf, wleaf); + +% Skeleton indices for each side +leftSkel = 1:nskel; +downSkel = nskel+1:2*nskel; +hypotSkel = 2*nskel+1:3*nskel; + +% Leaf indices for each side +leftLeaf = 1:nleaf; +%downLeaf = [1 nleaf+1:2*nleaf-1]; +downLeaf = [1 nleaf+1:2:3*nleaf-3]; +%hypotLeaf = [nleaf 2*nleaf:3*nleaf-3 2*nleaf-1]; +hypotLeaf = [nleaf:2:3*nleaf-4 3*nleaf-3]; + +P = zeros(3*nskel, 3*nleaf-3); +P(leftSkel, leftLeaf) = B; +P(downSkel, downLeaf) = B; +P(hypotSkel, hypotLeaf) = B; + +end diff --git a/+surfaceop/@leaf/initialize_ItI.m b/+surfaceop/@leaf/initialize_ItI_quad.m similarity index 95% rename from +surfaceop/@leaf/initialize_ItI.m rename to +surfaceop/@leaf/initialize_ItI_quad.m index a3bfa0e..f55cd4b 100644 --- a/+surfaceop/@leaf/initialize_ItI.m +++ b/+surfaceop/@leaf/initialize_ItI_quad.m @@ -1,4 +1,4 @@ -function L = initialize_ItI(op, dom, eta, rhs) +function L = initialize_ItI_quad(op, dom, eta, rhs) %INITIALIZE Initialize an array of LEAF objects. % L = SURFACEOP.LEAF.INITIALIZE(OP, DOM) returns a cell array L of LEAF % objects which contain the solution and D2N operators for Poisson's @@ -24,6 +24,7 @@ numPatches = length(dom); n = size(dom.x{1}, 1); +npts = n^2; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %%%%%%%%%%%%%%%%%%%%%%% DEFINE REFERENCE GRID %%%%%%%%%%%%%%%%%%%%%%%%%% @@ -67,9 +68,9 @@ % Compute binormal vectors [NL, NR, ND, NU] = binormals(dom); -ux = reshape([dom.ux{:}], [n^2 numPatches]); vx = reshape([dom.vx{:}], [n^2 numPatches]); -uy = reshape([dom.uy{:}], [n^2 numPatches]); vy = reshape([dom.vy{:}], [n^2 numPatches]); -uz = reshape([dom.uz{:}], [n^2 numPatches]); vz = reshape([dom.vz{:}], [n^2 numPatches]); +ux = reshape([dom.ux{:}], [npts numPatches]); vx = reshape([dom.vx{:}], [npts numPatches]); +uy = reshape([dom.uy{:}], [npts numPatches]); vy = reshape([dom.vy{:}], [npts numPatches]); +uz = reshape([dom.uz{:}], [npts numPatches]); vz = reshape([dom.vz{:}], [npts numPatches]); ItI_scl0 = {ones(nskel,1) ; ones(nskel,1) ; ones(nskel,1) ; ones(nskel,1)}; @@ -82,9 +83,9 @@ Du = kron(D, I); Dv = kron(I, D); -X = reshape([dom.x{:}], [n^2 numPatches]); -Y = reshape([dom.y{:}], [n^2 numPatches]); -Z = reshape([dom.z{:}], [n^2 numPatches]); +X = reshape([dom.x{:}], [npts numPatches]); +Y = reshape([dom.y{:}], [npts numPatches]); +Z = reshape([dom.z{:}], [npts numPatches]); flags = structfun(@(f) ~(isscalar(f) && isnumeric(f) && f==0), op, 'UniformOutput', false); @@ -93,7 +94,7 @@ if ( isa(op.(name), 'function_handle') ) op.(name) = feval(op.(name), X, Y, Z); elseif ( isa(op.(name), 'surfacefun') ) - op.(name) = reshape([op.(name).vals{:}], [n^2 numPatches]); + op.(name) = reshape([op.(name).vals{:}], [npts numPatches]); elseif ( isscalar(op.(name)) ) op.(name) = repmat(op.(name), [1 numPatches]); end @@ -107,7 +108,7 @@ rhs = feval(rhs, X(ii,:), Y(ii,:), Z(ii,:)); elseif ( isa(rhs, 'surfacefun') ) vals = rhs.vals; - rhs = reshape([vals{:}], [n^2 numPatches]); + rhs = reshape([vals{:}], [npts numPatches]); rhs = rhs(ii,:); elseif ( isnumeric(rhs) && isscalar(rhs) ) rhs = repmat(rhs, numIntPts, numPatches); @@ -131,7 +132,7 @@ x(1,1) y(1,1) z(1,1) x(1,n) y(1,n) z(1,n) nskel ; % "Down" side x(n,1) y(n,1) z(n,1) x(n,n) y(n,n) z(n,n) nskel ]; % "Up" side - A = zeros(n^2); + A = zeros(npts); Dx = ux(:,k).*Du + vx(:,k).*Dv; Dy = uy(:,k).*Du + vy(:,k).*Dv; Dz = uz(:,k).*Du + vz(:,k).*Dv; @@ -202,7 +203,7 @@ rightLeaf = 3*n-3:4*n-4; upLeaf = [n:2:3*n-4 4*n-4]; downLeaf = [1 n+1:2:3*n-3]; - bc = zeros(numBdyPts, n^2); + bc = zeros(numBdyPts, npts); bc(leftLeaf,:) = bc(leftLeaf,:) + bc_w; bc(rightLeaf,:) = bc(rightLeaf,:) + bc_e; bc(downLeaf,:) = bc(downLeaf,:) + bc_s; diff --git a/+surfaceop/@leaf/initialize_ItI_tri.m b/+surfaceop/@leaf/initialize_ItI_tri.m new file mode 100644 index 0000000..a403a5e --- /dev/null +++ b/+surfaceop/@leaf/initialize_ItI_tri.m @@ -0,0 +1,372 @@ +function L = initialize_ItI_tri(op, dom, eta, rhs) +%INITIALIZE Initialize an array of LEAF objects. +% L = SURFACEOP.LEAF.INITIALIZE(OP, DOM) returns a cell array L of LEAF +% objects which contain the solution and D2N operators for Poisson's +% equation on the domain DOM with zero righthand side. +% +% L = SURFACEOP.LEAF.INITIALIZE(OP, DOM, RHS) is as above, but with the +% righthand side RHS, which may be a scalar or a function handle. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %%%%%%%%%%%%%%%%%%%%%%%%%%% PARSE INPUTS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +if ( isempty(dom) ) + L = []; + return +end + +assert(isa(dom, 'surfacemesh'), 'Invalid domain.'); + +if ( nargin < 3 ) + % Default to homogeneous problem: + rhs = 0; +end + +numPatches = length(dom); +npts = length(dom.x{1}); +n = order(dom)+1; + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %%%%%%%%%%%%%%%%%%%%%%% DEFINE REFERENCE GRID %%%%%%%%%%%%%%%%%%%%%%%%%% + +[xx, yy] = trianglepts(n); +ee_left = 1:n; +ee_down = cumsum([1 (n:-1:2)]); +ee_hypot = cumsum([n (n-1:-1:1)]); +eeIdx = unique([ee_left ee_down ee_hypot]); +ee = false(size(xx)); +ee(eeIdx) = true; % Boundary indices +ii = ~ee; % Interior indices +numBdyPts = sum(ee(:)); +numIntPts = sum(ii(:)); + +% Impose the PDE on the interior nodes +ipde = ii; + +% Impose the BCs on the interior of the boundary nodes +% TODO +ibc = false(n, 1); +ibc(2:n-1) = true; + +% Skeleton mappings +nskel = n-2; +numSkelPts = 3*nskel; +S2L = skel2leaf(n, nskel); % Don't sparsify for speed +L2S = leaf2skel(nskel, n); +B2S = bdy2skel(nskel, n); % TODO +xskel = chebpts(nskel, 1); +[xleaf, ~, wleaf] = chebpts(n, 2); +B = barymat(xskel, xleaf, wleaf); +w = chebtech1.quadwts(nskel); w = w(:); +wskel = [0.5*w ; 0.5*w ; sqrt(2)/2*w]; + +% Skeleton indices for each side +leftSkel = 1:nskel; +downSkel = nskel+1:2*nskel; +hypotSkel = 2*nskel+1:3*nskel; + +% Compute binormal vectors +[NL, ND, NH] = binormals(dom); + +ux = reshape([dom.ux{:}], [npts numPatches]); vx = reshape([dom.vx{:}], [npts numPatches]); +uy = reshape([dom.uy{:}], [npts numPatches]); vy = reshape([dom.vy{:}], [npts numPatches]); +uz = reshape([dom.uz{:}], [npts numPatches]); vz = reshape([dom.vz{:}], [npts numPatches]); + +ItI_scl0 = {ones(nskel,1) ; ones(nskel,1) ; ones(nskel,1)}; + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %%%%%%%%%%%%%%%%%%%%%%%%% DEFINE OPERATORS %%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +[K, Ku, Kv] = koornwinder(n-1, xx, yy); +Du = Ku / K; +Dv = Kv / K; +II = eye(npts); + +X = reshape([dom.x{:}], [npts numPatches]); +Y = reshape([dom.y{:}], [npts numPatches]); +Z = reshape([dom.z{:}], [npts numPatches]); + +flags = structfun(@(f) ~(isscalar(f) && isnumeric(f) && f==0), op, 'UniformOutput', false); + +for name = fieldnames(op).' + name = name{1}; + if ( isa(op.(name), 'function_handle') ) + op.(name) = feval(op.(name), X, Y, Z); + elseif ( isa(op.(name), 'surfacefun') ) + op.(name) = reshape([op.(name).vals{:}], [npts numPatches]); + elseif ( isscalar(op.(name)) ) + op.(name) = repmat(op.(name), [1 numPatches]); + end +end + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %%%%%%%%%%%%%%%%%%%%%%%%%%% CONSTANT RHS? %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% Evaluate non-constant RHSs if required: +if ( isa(rhs, 'function_handle') ) + rhs = feval(rhs, X(ii,:), Y(ii,:), Z(ii,:)); +elseif ( isa(rhs, 'surfacefun') ) + vals = rhs.vals; + rhs = reshape([vals{:}], [npts numPatches]); + rhs = rhs(ii,:); +elseif ( isnumeric(rhs) && isscalar(rhs) ) + rhs = repmat(rhs, numIntPts, numPatches); +end + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %%%%%%%%%%%%%%%%%%%%%%% SOLVE LOCAL PROBLEMS %%%%%%%%%%%%%%%%%%%%%%%%%%% + +% Initialize +L = cell(numPatches, 1); + +% Loop over each patch: +for k = 1:numPatches + + % Define the left and right edges for this patch: + x = dom.x{k}; + y = dom.y{k}; + z = dom.z{k}; + edges = [ x(1) y(1) z(1) x(n) y(n) z(n) nskel ; % "Left" side + x(1) y(1) z(1) x(end) y(end) z(end) nskel ; % "Down" side + x(n) y(n) z(n) x(end) y(end) z(end) nskel ]; % "Hypot" side + + A = zeros(npts); + Dx = ux(:,k).*Du + vx(:,k).*Dv; + Dy = uy(:,k).*Du + vy(:,k).*Dv; + Dz = uz(:,k).*Du + vz(:,k).*Dv; + J = dom.J{k}(:); + + if ( dom.singular(k) ) + + % Assemble matrix: + if ( flags.dxx ), A = A + op.dxx(:,k).*(J.*(Dx*Dx)-(Dx*J).*Dx); end + if ( flags.dyy ), A = A + op.dyy(:,k).*(J.*(Dy*Dy)-(Dy*J).*Dy); end + if ( flags.dzz ), A = A + op.dzz(:,k).*(J.*(Dz*Dz)-(Dz*J).*Dz); end + if ( flags.dxy ), A = A + op.dxy(:,k).*(J.*(Dx*Dy)-(Dx*J).*Dy); end + if ( flags.dyx ), A = A + op.dyx(:,k).*(J.*(Dy*Dx)-(Dy*J).*Dx); end + if ( flags.dyz ), A = A + op.dyz(:,k).*(J.*(Dy*Dz)-(Dy*J).*Dz); end + if ( flags.dzy ), A = A + op.dzy(:,k).*(J.*(Dz*Dy)-(Dz*J).*Dy); end + if ( flags.dxz ), A = A + op.dxz(:,k).*(J.*(Dx*Dz)-(Dx*J).*Dz); end + if ( flags.dzx ), A = A + op.dzx(:,k).*(J.*(Dz*Dx)-(Dz*J).*Dx); end + if ( flags.dx ), A = A + op.dx(:,k).*J.^2.*Dx; end + if ( flags.dy ), A = A + op.dy(:,k).*J.^2.*Dy; end + if ( flags.dz ), A = A + op.dz(:,k).*J.^2.*Dz; end + if ( flags.b ), A = A + op.b(:,k).*J.^3.*II; end + + % Construct solution operator: + dA = decomposition(A(ii,ii), 'cod'); + Ainv = @(u) dA \ (J(ii).^3.*u); + S = dA \ ([-A(ii,ee), J(ii).^3.*rhs(:,k)]); + + dx = L2S * (J(ee).^2.*Dx(ee,:)); + dy = L2S * (J(ee).^2.*Dy(ee,:)); + dz = L2S * (J(ee).^2.*Dz(ee,:)); + + % The D2N map needs to be scaled on each side (e.g. when being + % merged) to account for the Jacobian scaling which has been + % factored out of the coordinate derivative maps. This scaling + % is not known until the merge stage, as it depends on the + % scaling of the neighboring patch. + Jss = L2S * J(ee).^3; + ItI_scl = {Jss(leftSkel); Jss(downSkel); Jss(hypotSkel)}; + + else + + % Assemble matrix: + if ( flags.dxx ), A = A + op.dxx(:,k).*(Dx*Dx); end + if ( flags.dyy ), A = A + op.dyy(:,k).*(Dy*Dy); end + if ( flags.dzz ), A = A + op.dzz(:,k).*(Dz*Dz); end + if ( flags.dxy ), A = A + op.dxy(:,k).*(Dx*Dy); end + if ( flags.dyx ), A = A + op.dyx(:,k).*(Dy*Dx); end + if ( flags.dyz ), A = A + op.dyz(:,k).*(Dy*Dz); end + if ( flags.dzy ), A = A + op.dzy(:,k).*(Dz*Dy); end + if ( flags.dxz ), A = A + op.dxz(:,k).*(Dx*Dz); end + if ( flags.dzx ), A = A + op.dzx(:,k).*(Dz*Dx); end + if ( flags.dx ), A = A + op.dx(:,k).*Dx; end + if ( flags.dy ), A = A + op.dy(:,k).*Dy; end + if ( flags.dz ), A = A + op.dz(:,k).*Dz; end + if ( flags.b ), A = A + op.b(:,k).*II; end + + dir_left = II(ee_left,:); neu_left = NL(:,1,k).*Dx(ee_left,:) + NL(:,2,k).*Dy(ee_left,:) + NL(:,3,k).*Dz(ee_left,:); + dir_down = II(ee_down,:); neu_down = ND(:,1,k).*Dx(ee_down,:) + ND(:,2,k).*Dy(ee_down,:) + ND(:,3,k).*Dz(ee_down,:); + dir_hypot = II(ee_hypot,:); neu_hypot = NH(:,1,k).*Dx(ee_hypot,:) + NH(:,2,k).*Dy(ee_hypot,:) + NH(:,3,k).*Dz(ee_hypot,:); + + bc_left = neu_left + 1i*eta*dir_left; + bc_down = neu_down + 1i*eta*dir_down; + bc_hypot = neu_hypot + 1i*eta*dir_hypot; + + leftLeaf = 1:n; + downLeaf = [1 n+1:2:3*n-3]; + hypotLeaf = [n:2:3*n-4 3*n-3]; + corners = [1 n 3*n-3]; + + bc = zeros(numBdyPts, npts); + bc(leftLeaf,:) = bc(leftLeaf,:) + bc_left; + bc(downLeaf,:) = bc(downLeaf,:) + bc_down; + bc(hypotLeaf,:) = bc(hypotLeaf,:) + bc_hypot; + bc(corners,:) = bc(corners,:)/2; + + BC = [ zeros(numIntPts, numBdyPts) ; eye(numBdyPts) ]; + RHS = [rhs(:,k); zeros(numBdyPts, 1)]; + + LL = [ A(ipde,:) ; bc ]; + dA = matlab.internal.decomposition.DenseLU(LL); + Ainv = @(u) solve(dA, u, false); + S = Ainv([BC RHS]); + + ItI_scl = ItI_scl0; + end + + % Append boundary points to solution operator and extract the + % particular solution to store separately: + u_part = S(:,end); + S = S(:,1:end-1) * S2L; + + % Construct outgoing impedance operator: + normal_d = [ neu_left - 1i*eta*dir_left ; + neu_down - 1i*eta*dir_down ; + neu_hypot - 1i*eta*dir_hypot ]; + normal_d = B2S * normal_d; + + % Construct the ItI map and particular flux: + ItI = normal_d * S; + du_part = normal_d * u_part; + + JJ = L2S * sqrt(J(ee)); + ww = wskel .* JJ; + xyz = L2S * [x(ee) y(ee) z(ee)]; + + % Assemble the patch: + L{k} = surfaceop.leaf(dom, n, k, S, ItI, ItI_scl, u_part, du_part, edges, xyz, ww, Ainv, normal_d); + +end + +end + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %%%%%%%%%%%%%%%%%%%%%%%%%% DEFINE OPERATORS %%%%%%%%%%%%%%%%%%%%%%%%%%%% + +function [nl, nd, nh] = binormals(dom) +%BINORMALS Compute the binormal vectors for a surfacemesh. + +n = order(dom)+1; +eleft = 1:n; +edown = cumsum([1 (n:-1:2)]); +ehypot = cumsum([n (n-1:-1:1)]); + +npts = size(dom.x{1}, 1); +sz = [npts 1 length(dom)]; +xu = reshape([dom.xu{:}], sz); xv = reshape([dom.xv{:}], sz); +yu = reshape([dom.yu{:}], sz); yv = reshape([dom.yv{:}], sz); +zu = reshape([dom.zu{:}], sz); zv = reshape([dom.zv{:}], sz); + +% Normal vectors to the surface (unnormalized) +nl = -[xu(eleft,:,:) yu(eleft,:,:) zu(eleft,:,:)]; +nd = -[xv(edown,:,:) yv(edown,:,:) zv(edown,:,:)]; +nh = [xu(ehypot,:,:) yu(ehypot,:,:) zu(ehypot,:,:)] + ... + [xv(ehypot,:,:) yv(ehypot,:,:) zv(ehypot,:,:)]; + +% Tangent vectors to the element boundary (normalized) +tl = normalize([xv(eleft,:,:) yv(eleft,:,:) zv(eleft,:,:)]); +td = normalize([xu(edown,:,:) yu(edown,:,:) zu(edown,:,:)]); +th = normalize( -[xv(ehypot,:,:) yv(ehypot,:,:) zv(ehypot,:,:)] + ... + [xu(ehypot,:,:) yu(ehypot,:,:) zu(ehypot,:,:)] ); + +% Binormal vectors (normalized) +nl = normalize(nl - tl .* sum(nl.*tl, 2)); +nd = normalize(nd - td .* sum(nd.*td, 2)); +nh = normalize(nh - th .* sum(nh.*th, 2)); + +end + +function v = normalize(v) + +v = v ./ sqrt(v(:,1,:).^2 + v(:,2,:).^2 + v(:,3,:).^2); + +end + +function P = skel2leaf(nleaf, nskel) +%SKEL2LEAF Boundary interpolation matrix. +% SKEL2LEAF(NLEAF, NSKEL) returns the (3*NLEAF-3) x 3*NSKEL matrix that +% maps 3 pieces of length-NSKEL first-kind boundary values to 3*NLEAF-3 +% second-kind boundary values, including the corners. At each corner, the +% average of the two interpolated values is used. + +[xskel, ~, wskel] = chebpts(nskel, 1); +[xleaf, ~, wleaf] = chebpts(nleaf, 2); +B = barymat(xleaf, xskel, wskel); + +% Skeleton indices for each side +leftSkel = 1:nskel; +downSkel = nskel+1:2*nskel; +hypotSkel = 2*nskel+1:3*nskel; + +% Leaf indices for each side +leftLeaf = 1:nleaf; +downLeaf = [1 nleaf+1:2:3*nleaf-3]; +hypotLeaf = [nleaf:2:3*nleaf-4 3*nleaf-3]; + +P = zeros(3*nleaf-3, 3*nskel); +P(leftLeaf, leftSkel) = B; +P(downLeaf, downSkel) = B; +P(hypotLeaf, hypotSkel) = B; + +% Average the corners: +corners = [1 nleaf 3*nleaf-3]; +P(corners,:) = P(corners,:)/2; + +end + +function P = leaf2skel(nskel, nleaf) +%LEAF2SKEL Boundary interpolation matrix. +% LEAF2SKEL(NSKEL, NLEAF) returns the 3*NSKEL x (3*NLEAF-3) matrix that +% maps 3*NLEAF-3 second-kind boundary values to 3 pieces of length-NSKEL +% first-kind boundary values. + +[xskel, ~, wskel] = chebpts(nskel, 1); +[xleaf, ~, wleaf] = chebpts(nleaf, 2); +B = barymat(xskel, xleaf, wleaf); + +% Skeleton indices for each side +leftSkel = 1:nskel; +downSkel = nskel+1:2*nskel; +hypotSkel = 2*nskel+1:3*nskel; + +% Leaf indices for each side +leftLeaf = 1:nleaf; +downLeaf = [1 nleaf+1:2:3*nleaf-3]; +hypotLeaf = [nleaf:2:3*nleaf-4 3*nleaf-3]; + +P = zeros(3*nskel, 3*nleaf-3); +P(leftSkel, leftLeaf) = B; +P(downSkel, downLeaf) = B; +P(hypotSkel, hypotLeaf) = B; + +end + +function P = bdy2skel(nskel, nleaf) +%BDY2SKEL Boundary interpolation matrix. +% BDY2SKEL(NSKEL, NLEAF) returns the 3*NSKEL x 3*NLEAF matrix that maps +% 3 pieces of length-NLEAF second-kind boundary values to 3 pieces of +% length-NSKEL first-kind boundary values. + +[xskel, ~, wskel] = chebpts(nskel, 1); +[xleaf, ~, wleaf] = chebpts(nleaf, 2); +B = barymat(xskel, xleaf, wleaf); + +% Skeleton indices for each side +leftSkel = 1:nskel; +downSkel = nskel+1:2*nskel; +hypotSkel = 2*nskel+1:3*nskel; + +% Leaf boundary indices for each side +leftBdy = 1:nleaf; +downBdy = nleaf+1:2*nleaf; +hypotBdy = 2*nleaf+1:3*nleaf; + +P = zeros(3*nskel, 3*nleaf); +P(leftSkel, leftBdy) = B; +P(downSkel, downBdy) = B; +P(hypotSkel, hypotBdy) = B; + +end diff --git a/+surfaceop/@leaf/leaf.m b/+surfaceop/@leaf/leaf.m index 92e9cb4..d4d6c26 100644 --- a/+surfaceop/@leaf/leaf.m +++ b/+surfaceop/@leaf/leaf.m @@ -50,8 +50,10 @@ methods ( Static ) % Initialize an array of LEAF objects. - P = initialize_DtN(op, dom, rhs); - P = initialize_ItI(op, dom, eta, rhs); + P = initialize_DtN_quad(op, dom, rhs); + P = initialize_DtN_tri(op, dom, rhs); + P = initialize_ItI_quad(op, dom, eta, rhs); + P = initialize_ItI_tri(op, dom, eta, rhs); end diff --git a/+surfaceop/@leaf/solve_DtN.m b/+surfaceop/@leaf/solve_DtN.m index ae66e83..583199c 100644 --- a/+surfaceop/@leaf/solve_DtN.m +++ b/+surfaceop/@leaf/solve_DtN.m @@ -4,7 +4,7 @@ % Extract the domain from the patch: id = P.id; -n = size(P.domain.x{id}, 1); +[nv, nu] = size(P.domain.x{id}); if ( ~isnumeric(bc) ) % Evaluate the RHS if given a function handle: @@ -20,7 +20,7 @@ % Return cell output for consistency with PARENT/SOLVE(): U = cell(1, size(u, 2)); for k = 1:size(u, 2) - U{k} = reshape(u(:,k), n, n); + U{k} = reshape(u(:,k), nv, nu); end u = U; diff --git a/+surfaceop/@leaf/solve_ItI.m b/+surfaceop/@leaf/solve_ItI.m index 195da07..d3cf805 100644 --- a/+surfaceop/@leaf/solve_ItI.m +++ b/+surfaceop/@leaf/solve_ItI.m @@ -4,7 +4,7 @@ % Extract the domain from the patch: id = P.id; -n = size(P.domain.x{id}, 1); +[nv, nu] = size(P.domain.x{id}); if ( ~isnumeric(bc) ) % Evaluate the RHS if given a function handle: @@ -20,7 +20,7 @@ % Return cell output for consistency with PARENT/SOLVE(): U = cell(1, size(u, 2)); for k = 1:size(u, 2) - U{k} = reshape(u(:,k), n, n); + U{k} = reshape(u(:,k), nv, nu); end u = U; diff --git a/+surfaceop/@leaf/updateRHS_DtN.m b/+surfaceop/@leaf/updateRHS_DtN.m index 9fca18a..6986189 100644 --- a/+surfaceop/@leaf/updateRHS_DtN.m +++ b/+surfaceop/@leaf/updateRHS_DtN.m @@ -16,28 +16,39 @@ dom = P.domain; id = P.id; -nrhs = 1; -ii = false(n); -ii(2:n-1,2:n-1) = true; +nrhs = size(rhs, 2); +numLeafNodes = numel(dom.x{id}); +switch ( dom.ptype(id) ) + case 'quad' + ii = false(n); + ii(2:n-1,2:n-1) = true; + case 'tri' + eleft = 1:n; + edown = cumsum([1 (n:-1:2)]); + ehypot = cumsum([n (n-1:-1:1)]); + eeIdx = unique([eleft edown ehypot]); + ee = false(numLeafNodes, 1); + ee(eeIdx) = true; + ii = ~ee; +end if ( iscell(rhs) && isa(rhs{1}, 'function_handle') ) rhs = rhs{1}; end if ( iscell(rhs) ) - nrhs = size(rhs, 2); - rhs = reshape([rhs{1,:}], n^2, nrhs); + rhs = reshape([rhs{1,:}], numLeafNodes, nrhs); end % Define scalar RHSs: if ( isnumeric(rhs) && isscalar(rhs) ) % Constant RHS. - rhs = repmat(rhs, n^2, 1); + rhs = repmat(rhs, numLeafNodes, 1); elseif ( isnumeric(rhs) && ~isscalar(rhs) ) % We already have the values of the RHS. elseif ( isa(rhs, 'function_handle') ) rhs = feval(rhs, dom.x{id}, dom.y{id}, dom.z{id}); - rhs = reshape(rhs, n^2, 1); + rhs = reshape(rhs, numLeafNodes, 1); end % Restrict to interior nodes. @@ -56,12 +67,12 @@ S = P.Ainv(rhs); % Append boundary points to solution operator: -tmpS = zeros(n^2, nrhs); +tmpS = zeros(numLeafNodes, nrhs); tmpS(ii,:) = S; S = tmpS; P.u_part = S; % Normal derivative: -P.du_part = P.normal_d * S; +P.du_part = P.normal_d * P.u_part; end diff --git a/+surfaceop/@leaf/updateRHS_ItI.m b/+surfaceop/@leaf/updateRHS_ItI.m index af91c4c..bc02550 100644 --- a/+surfaceop/@leaf/updateRHS_ItI.m +++ b/+surfaceop/@leaf/updateRHS_ItI.m @@ -17,26 +17,38 @@ id = P.id; nrhs = size(rhs, 2); -ii = false(n); -ii(2:n-1,2:n-1) = true; +numLeafNodes = numel(dom.x{id}); +switch ( dom.ptype(id) ) + case 'quad' + ii = false(n); + ii(2:n-1,2:n-1) = true; + case 'tri' + eleft = 1:n; + edown = cumsum([1 (n:-1:2)]); + ehypot = cumsum([n (n-1:-1:1)]); + eeIdx = unique([eleft edown ehypot]); + ee = false(numLeafNodes, 1); + ee(eeIdx) = true; + ii = ~ee; +end if ( iscell(rhs) && isa(rhs{1}, 'function_handle') ) rhs = rhs{1}; end if ( iscell(rhs) ) - rhs = reshape([rhs{1,:}], n^2, nrhs); + rhs = reshape([rhs{1,:}], numLeafNodes, nrhs); end % Define scalar RHSs: if ( isnumeric(rhs) && isscalar(rhs) ) % Constant RHS. - rhs = repmat(rhs, n^2, 1); + rhs = repmat(rhs, numLeafNodes, 1); elseif ( isnumeric(rhs) && ~isscalar(rhs) ) % We already have the values of the RHS. elseif ( isa(rhs, 'function_handle') ) rhs = feval(rhs, dom.x{id}, dom.y{id}, dom.z{id}); - rhs = reshape(rhs, n^2, 1); + rhs = reshape(rhs, numLeafNodes, 1); end % Restrict to interior nodes. @@ -52,7 +64,8 @@ error('SURFACEOP:LEAF:updateRHS:operatorNotStored', ... 'Discretized operator A was not stored. Cannot update RHS.'); end -P.u_part = P.Ainv([rhs ; zeros(4*n-4, nrhs)]); +numBdyPts = sum(~ii(:)); +P.u_part = P.Ainv([rhs ; zeros(numBdyPts, nrhs)]); % Normal derivative: P.du_part = P.normal_d * P.u_part; diff --git a/+surfaceop/@parent/solve_DtN.m b/+surfaceop/@parent/solve_DtN.m index 2bd1eed..0b21135 100644 --- a/+surfaceop/@parent/solve_DtN.m +++ b/+surfaceop/@parent/solve_DtN.m @@ -32,9 +32,9 @@ idx2 = cat(1, P.idx2{:}); % idx2 = cell2mat(P.idx2.'); % Assemble boundary conditions for child patches: -ubc1 = ones(size(P.child1.S, 2)-1, size(P.u_part, 2)); +ubc1 = ones(size(P.child1.S, 2), size(P.u_part, 2)); ubc1(idx1,:) = [bc(i1,:) ; P.flip1.'*u]; -ubc2 = ones(size(P.child2.S, 2)-1, size(P.u_part, 2)); +ubc2 = ones(size(P.child2.S, 2), size(P.u_part, 2)); ubc2(idx2,:) = [bc(i2,:) ; P.flip2.'*u]; % Solve for the child patches: diff --git a/+surfaceop/@parent/solve_ItI.m b/+surfaceop/@parent/solve_ItI.m index a09c812..30bad11 100644 --- a/+surfaceop/@parent/solve_ItI.m +++ b/+surfaceop/@parent/solve_ItI.m @@ -34,9 +34,9 @@ idx2 = cat(1, P.idx2{:}); % idx2 = cell2mat(P.idx2.'); % Assemble boundary conditions for child patches: -ubc1 = ones(size(P.child1.xyz, 1)-1, size(P.u_part{1}, 2)); +ubc1 = ones(size(P.child1.xyz, 1), size(P.u_part{1}, 2)); ubc1(idx1,:) = [bc(i1,:) ; P.flip1.'*ua]; -ubc2 = ones(size(P.child2.xyz, 1)-1, size(P.u_part{2}, 2)); +ubc2 = ones(size(P.child2.xyz, 1), size(P.u_part{2}, 2)); ubc2(idx2,:) = [bc(i2,:) ; P.flip2.'*ub]; % Solve for the child patches: diff --git a/+surfaceop/@patch/intersect.m b/+surfaceop/@patch/intersect.m index c26ea6d..3d2d264 100644 --- a/+surfaceop/@patch/intersect.m +++ b/+surfaceop/@patch/intersect.m @@ -29,18 +29,12 @@ edgesA = a.edges; edgesB = b.edges; -sclx = 0; -scly = 0; -sclz = 0; -for k = 1:4 - sclx = max(sclx, abs(edgesA(k,4) - edgesA(k,1))); - sclx = max(sclx, abs(edgesB(k,4) - edgesB(k,1))); - scly = max(scly, abs(edgesA(k,5) - edgesA(k,2))); - scly = max(scly, abs(edgesB(k,5) - edgesB(k,2))); - sclz = max(sclz, abs(edgesA(k,6) - edgesA(k,3))); - sclz = max(sclz, abs(edgesB(k,6) - edgesB(k,3))); -end -scl = max([sclx scly sclz]); +scl = max([ abs(edgesA(:,4) - edgesA(:,1)) ; + abs(edgesA(:,5) - edgesA(:,2)) ; + abs(edgesA(:,6) - edgesA(:,3)) ; + abs(edgesB(:,4) - edgesB(:,1)) ; + abs(edgesB(:,5) - edgesB(:,2)) ; + abs(edgesB(:,6) - edgesB(:,3)) ]); % Check for intersecting edges (with a tolerance): tol = 1e-8 * scl; diff --git a/@surfaceop/initialize.m b/@surfaceop/initialize.m index a8e111a..da9a8af 100644 --- a/@surfaceop/initialize.m +++ b/@surfaceop/initialize.m @@ -17,9 +17,19 @@ function initialize(S, varargin) % Initialize all leaf patches: switch S.method case 'DtN' - S.patches = surfaceop.leaf.initialize_DtN(S.op, S.domain, varargin{:}); + switch ( S.domain.ptype(1) ) + case 'tri' + S.patches = surfaceop.leaf.initialize_DtN_tri(S.op, S.domain, varargin{:}); + case 'quad' + S.patches = surfaceop.leaf.initialize_DtN_quad(S.op, S.domain, varargin{:}); + end case 'ItI' - S.patches = surfaceop.leaf.initialize_ItI(S.op, S.domain, S.eta, varargin{:}); + switch ( S.domain.ptype(1) ) + case 'tri' + S.patches = surfaceop.leaf.initialize_ItI_tri(S.op, S.domain, S.eta, varargin{:}); + case 'quad' + S.patches = surfaceop.leaf.initialize_ItI_quad(S.op, S.domain, S.eta, varargin{:}); + end end end From 9b97088553639973e55e8a95ea0d1938b5a82632 Mon Sep 17 00:00:00 2001 From: Dan Fortunato Date: Wed, 29 Oct 2025 11:44:52 -0400 Subject: [PATCH 6/7] Convert between quad and triangle meshes --- @surfacemesh/quad2tri.m | 80 +++++++++++++++++++++++++++++++++++++++++ 1 file changed, 80 insertions(+) create mode 100644 @surfacemesh/quad2tri.m diff --git a/@surfacemesh/quad2tri.m b/@surfacemesh/quad2tri.m new file mode 100644 index 0000000..833ad13 --- /dev/null +++ b/@surfacemesh/quad2tri.m @@ -0,0 +1,80 @@ +function domnew = quad2tri(dom) +%QUAD2TRI Convert a quadrilateral mesh to a triangle mesh. + +triIdx = ( dom.ptype == surfacemesh.patchtype.tri ); +quadIdx = ( dom.ptype == surfacemesh.patchtype.quad ); + +ntri = sum(triIdx); +nquad = sum(quadIdx); + +xnew = cell(ntri + 2*nquad, 1); +ynew = cell(ntri + 2*nquad, 1); +znew = cell(ntri + 2*nquad, 1); + +x = dom.x; +y = dom.y; +z = dom.z; + +if ( quadIdx(1) ) + n = size(x{1}, 1); +else + npts = length(x{1}); + n = (sqrt(8*npts+1)-1) / 2; +end + +[xtri, ytri] = trianglepts(n, domain='biunit'); +V2C = chebtech2.vals2coeffs(eye(n)); +V2C = kron(V2C, V2C); +T1 = chebpoly2(n, xtri, ytri); +B1 = T1 * V2C; +T2 = chebpoly2(n, -xtri, -ytri); +B2 = T2 * V2C; + +knew = 1; +for k = 1:length(dom) + if ( triIdx(k) ) + xnew(knew) = x(k); + ynew(knew) = y(k); + znew(knew) = z(k); + knew = knew+1; + else + xnew{knew} = B1*x{k}(:); + ynew{knew} = B1*y{k}(:); + znew{knew} = B1*z{k}(:); + xnew{knew+1} = B2*x{k}(:); + ynew{knew+1} = B2*y{k}(:); + znew{knew+1} = B2*z{k}(:); + knew = knew+2; + end +end + +domnew = surfacemesh(xnew, ynew, znew, 'tri'); + +end + +function V = chebpoly2(n, x, y) + +x = x(:); +y = y(:); +basis1d = chebpoly(0:n-1); +xbasis = basis1d(x); +ybasis = basis1d(y); +V = zeros(length(x), n^2); +for i = 1:n + for j = 1:n + V(:,(i-1)*n+j) = xbasis(:,i) .* ybasis(:,j); + end +end + +end + +function vals = bary2d(chebvals, x, y) + +vals = 0*x; +C = chebtech2.vals2coeffs( chebtech2.vals2coeffs( chebvals ).' ).'; +Cy = chebtech2.clenshaw(y(:), C).'; +for k = 1:numel(x) + vals(k) = chebtech2.clenshaw(x(k), Cy(:,k)); +end + +end From 0a7425ba58dc63663a49c9c50066f1ea0886fe5c Mon Sep 17 00:00:00 2001 From: Dan Fortunato Date: Wed, 29 Oct 2025 11:46:28 -0400 Subject: [PATCH 7/7] Add quad/tri sphere meshes for testing --- models/sphere.geo | 16 + models/sphere_quad.msh | 60350 +++++++++++++++++++++++++++++++++++++++ models/sphere_tri.msh | 32770 +++++++++++++++++++++ 3 files changed, 93136 insertions(+) create mode 100644 models/sphere.geo create mode 100644 models/sphere_quad.msh create mode 100644 models/sphere_tri.msh diff --git a/models/sphere.geo b/models/sphere.geo new file mode 100644 index 0000000..dbfbc59 --- /dev/null +++ b/models/sphere.geo @@ -0,0 +1,16 @@ +SetFactory("OpenCASCADE"); +Sphere(1) = {0, 0, 0, 1}; + +Mesh.ElementOrder = 10; +Mesh.HighOrderOptimize = 4; +Mesh.MeshSizeFactor = 1; + +Mesh 2; +Save "sphere_tri.msh"; +Delete Meshes; + +Mesh.RecombineAll = 1; +Mesh.RecombinationAlgorithm = 1; +Mesh.MeshSizeFactor = 0.7; +Mesh 2; +Save "sphere_quad.msh"; diff --git a/models/sphere_quad.msh b/models/sphere_quad.msh new file mode 100644 index 0000000..6d9d08a --- /dev/null +++ b/models/sphere_quad.msh @@ -0,0 +1,60350 @@ +$MeshFormat +4.1 0 8 +$EndMeshFormat +$Entities +2 3 1 1 +1 6.123233995736766e-17 -1.499759782661858e-32 1 0 +2 6.123233995736766e-17 -1.499759782661858e-32 -1 0 +1 -1.000000000616482e-07 -1.000000000615789e-07 0.9999999000000001 1.000000000616482e-07 1.000000000615789e-07 1.0000001 0 2 1 -1 +2 -1.000000002249202e-07 -1.000000002449294e-07 -1.0000001 1.0000001 1e-07 1.0000001 0 2 2 -1 +3 -1.000000000616482e-07 -1.000000000615789e-07 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