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note: this passage serves for the analysis of Alec Jacobson’s thesis
According to (2.50), mass matrix is given by:
function M = massmatrix(V,F, type) % MASSMATRIX mass matrix for the mesh given by V and F % % M = massmatrix(V,F, type) % % Inputs: % V #V x 3 matrix of vertex coordinates % F #F x 3 matrix of indices of triangle corners % type string containing type of mass matrix to compute % 'full': full mass matrix for p.w. linear fem % 'barycentric': diagonal lumped mass matrix obtained by summing 1/3 % 'voronoi': true voronoi area, except in cases where triangle is obtuse % then uses 1/2, 1/4, 1/4 % Output: % M #V by #V sparse mass matrix % % Copyright 2011, Alec Jacobson (jacobson@inf.ethz.ch) % % See also: massmatrix3 % % should change code below, so we don't need this transpose if(size(F,1) == 3) warning('F seems to be 3 by #F, it should be #F by 3'); end F = F'; % renaming indices of vertices of triangles for convenience i1 = F(1,:); i2 = F(2,:); i3 = F(3,:); %#F x 3 matrices of triangle edge vectors, named after opposite vertices v1 = V(i3,:) - V(i2,:); v2 = V(i1,:) - V(i3,:); v3 = V(i2,:) - V(i1,:); % computing the areas if size(V,2) == 2 % 2d vertex data dblA = v1(:,1).*v2(:,2)-v1(:,2).*v2(:,1); elseif size(V,2) == 3 %n = cross(v1,v2,2); dblA = multinorm(n,2); n = cross(v1,v2,2); % dblA = norm(n,2); % This does correct l2 norm of rows dblA = (sqrt(sum((n').^2)))'; else error('unsupported vertex dimension %d', size(V,2)) end if strcmp(type,'full') % arrays for matrix assembly using 'sparse' % indices and values of the element mass matrix entries in the order % (1,2), (2,1),(2,3), (3,2), (3,1), (1,3) (1,1), (2,2), (3,3); i = [i1 i2 i2 i3 i3 i1 i1 i2 i3]; j = [i2 i1 i3 i2 i1 i3 i1 i2 i3]; offd_v = dblA/24.; diag_v = dblA/12.; v = [offd_v,offd_v, offd_v,offd_v, offd_v,offd_v, diag_v,diag_v,diag_v]; M = sparse(i,j,v,size(V,1), size(V,1)); %seamanj: 根据Quadrature rules, 对角线上为A_T/6,非对角线上为A_T/12① %注意这里dblA为双倍的三角形面积 elseif strcmp(type,'barycentric') % only diagonal elements i = [i1 i2 i3]; j = [i1 i2 i3]; diag_v = dblA/6.; v = [diag_v,diag_v,diag_v]; M = sparse(i,j,v,size(V,1), size(V,1)); %the entry M^d_i is the one third the sum of the areas of incident %triangles on vertex i.② elseif strcmp(type,'voronoi') % just ported version of intrinsic code % edges numbered same as opposite vertices FT = F'; l = [ ... sqrt(sum((V(FT(:,2),:)-V(FT(:,3),:)).^2,2)) ... sqrt(sum((V(FT(:,3),:)-V(FT(:,1),:)).^2,2)) ... sqrt(sum((V(FT(:,1),:)-V(FT(:,2),:)).^2,2)) ... ]; % 求三角形的边长 M = massmatrix_intrinsic(l,F',size(V,1),'voronoi'); %The voronoi mass matrix entry M^d_i for vertex i is the sum of its %corresponding quadrilaterals from all incident triangles. %具体请参照下一个文件③ else error('bad mass matrix type') end % warn if any rows are all zero (probably unreferenced vertices) if(any(sum(M,2) == 0)) warning('Some rows have all zeros... probably unreferenced vertices..'); endend
①
注意这里它采用的是second set of quadrature rules for triangular elements
[②
③function [M] = massmatrix_intrinsic(l,F,nvert,masstype) % MASSMATRIX_INTRINSIC compute the mass matrix from edge lengths only % % [M] = massmatrix_intrinsic(l,F) % % Inputs: % l: #F by 3, array of edge lengths of edges opposite each face in F % F: #F by 3, list of indices of triangle corners % nvert: number of vertices, only needed to set size % masstype: full, barycentric, or voronoi % TODO: this is almost identical to massmatrix, % only the area computation is different, need to refactor % % here's a handy line to view mass matrix entries on plot: % text(UV(:,1), UV(:,2),zeros(size(UV,1),1),num2str(M(M>0))) % % Copyright 2011, Alec Jacobson (jacobson@inf.ethz.ch) % % See also: massmatrix % % should change code below, so we don't need this transpose if(size(F,1) == 3) warning('F seems to be 3 by #F, it should be #F by 3'); end F = F'; % renaming indices of vertices of triangles for convenience l1 = l(:,1); l2 = l(:,2); l3 = l(:,3); % semiperimeters s = (l1 + l2 + l3)*0.5; % Heron's formula for area dblA = 2*sqrt( s.*(s-l1).*(s-l2).*(s-l3)); % renaming indices of vertices of triangles for convenience i1 = F(1,:); i2 = F(2,:); i3 = F(3,:); if strcmp(masstype,'full') % arrays for matrix assembly using 'sparse' % indices and values of the element mass matrix entries in the order % (1,2), (2,1),(2,3), (3,2), (3,1), (1,3) (1,1), (2,2), (3,3); i = [i1 i2 i2 i3 i3 i1 i1 i2 i3]; j = [i2 i1 i3 i2 i1 i3 i1 i2 i3]; offd_v = dblA/24.; diag_v = dblA/12.; v = [offd_v,offd_v, offd_v,offd_v, offd_v,offd_v, diag_v,diag_v,diag_v]; elseif strcmp(masstype,'barycentric') % only diagonal elements i = [i1 i2 i3]; j = [i1 i2 i3]; diag_v = dblA/6.; v = [diag_v,diag_v,diag_v]; elseif strcmp(masstype,'voronoi') cosines = [ ... (l(:,3).^2+l(:,2).^2-l(:,1).^2)./(2*l(:,2).*l(:,3)), ... (l(:,1).^2+l(:,3).^2-l(:,2).^2)./(2*l(:,1).*l(:,3)), ... (l(:,1).^2+l(:,2).^2-l(:,3).^2)./(2*l(:,1).*l(:,2))]; %seamanj:求cosine barycentric = cosines.*l; %seamanj:求质心,质心坐标为 a^2(b^2+c^2-a^2),b^2(c^2+a^2-b^2),c^2(a^2+b^2-c^2)参见④ normalized_barycentric = barycentric./[sum(barycentric')' sum(barycentric')' sum(barycentric')']; %seamanj:Barycentric coordinates are homogeneous, so(t_1,t_2,t_3)=(ut_1,ut_2,ut_3) 引用④ areas = 0.25*sqrt( ... (l(:,1) + l(:,2) - l(:,3)).* ... (l(:,1) - l(:,2) + l(:,3)).* ... (-l(:,1) + l(:,2) + l(:,3)).* ... (l(:,1) + l(:,2) + l(:,3))); partial_triangle_areas = normalized_barycentric.*[areas areas areas]; %seamanj:the areas of the triangles ΔA_1A_2P, ΔA_1A_3P, and ΔA_2A_3P are proportional to the barycentric coordinates t_3, t_2, and t_1 of P 引用④ quads = [ (partial_triangle_areas(:,2)+ partial_triangle_areas(:,3))*0.5 ... (partial_triangle_areas(:,1)+ partial_triangle_areas(:,3))*0.5 ... (partial_triangle_areas(:,1)+ partial_triangle_areas(:,2))*0.5]; %seamanj:这里条件cosines(:,1)<0当筛选器,注意左右两边都筛选quads(cosines(:,1)<0,:) = [areas(cosines(:,1)<0,:)*0.5, ... areas(cosines(:,1)<0,:)*0.25, areas(cosines(:,1)<0,:)*0.25];quads(cosines(:,2)<0,:) = [areas(cosines(:,2)<0,:)*0.25, ... areas(cosines(:,2)<0,:)*0.5, areas(cosines(:,2)<0,:)*0.25];quads(cosines(:,3)<0,:) = [areas(cosines(:,3)<0,:)*0.25, ... areas(cosines(:,3)<0,:)*0.25, areas(cosines(:,3)<0,:)*0.5]; i = [i1 i2 i3]; j = [i1 i2 i3]; v = reshape(quads,size(quads,1)*3,1); else error('bad mass matrix type') end M = sparse(i,j,v,nvert, nvert); %这里会做叠加的事end
④
[ 大部分的内容图片已经给出,这里我想说明下为什么质心坐标的公式为 barycentric = cosines.*l; 注意质心到三角形三个顶点的距离相等,所以这里质心也是外接圆的球心 即这里我们看到的 而matlab里面为cosines.*l, 这里我想推导下 将cosines写开来: [(b2+c2−a2)/2bc,(c2+a2−b2)/2ac,(a2+b2−c2)/2ab] 将l写开来: [a,b,c] 那么cosines.*l则为 [(b2+c2−a2)∗a/2bc,(c2+a2−b2)∗b/2ac,(a2+b2−c2)∗c/2ab] 再根据 我们将cosines.*l里面的三个分量同时乘以 2abc ,则得到 [(b2+c2−a2)∗a2,(c2+a2−b2)∗b2,(a2+b2−c2)∗c2]