solve mass matrix in matlab

note: this passage serves for the analysis of Alec Jacobson’s thesis

1. what’s mass matrix

According to (2.50), mass matrix is given by:

Mij=∫ΩϕiϕjdA

2. matlab code analysis

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 ([e-mail])  %  % 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

[http://math2.uncc.edu/~shaodeng/TEACHING/math5172/Lectures/Lect_15.PDF]

②


③

function [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 ([e-mail])  %  % 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

[http://mathworld.wolfram.com/BarycentricCoordinates.html]
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]