solve stiffness matrix in matlab

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

1.what’s stiffness matrix

According to (2.16), stiffness matrix is given by:

Lij=∫Ω−∇ϕi∇ϕjdA

2. matlab code analysis

function L = cotmatrix(V,F)
  % COTMATRIX computes cotangent matrix (laplacian mesh operator), (mass/area
  % terms already cancelled out)
  %
  % L = cotmatrix(V,F)
  %
  % Inputs:
  % V #V x 3 matrix of vertex coordinates
  % F #F x 3 matrix of indices of triangle corners
  % Outputs:
  % L sparse #V x #V matrix of cot weights 
  %
  % Copyright 2011, Alec Jacobson ([email protected]), Denis Zorin
  %
  % See also: cotmatrix3
  %

  % 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,:); 
  %i1 = i1'; i2 = i2'; i3 = i3';
  % #F x 3 matrices of triangle edge vectors, named after opposite vertices
  %seamanj:这里i1,i2,i3是索引，应该为一个向量，无论列向量，行向量都行
  v1 = V(i3,:) - V(i2,:);  v2 = V(i1,:) - V(i3,:); v3 = V(i2,:) - V(i1,:);
  %seamanj:这里v1,v2,v3,对于某个特定索引来说是个行向量，然后多个索引从垂直方向延伸
  % computing areas 
  if size(V,2) == 2
      % 2d vertex data
      dblA = v1(:,1).*v2(:,2)-v1(:,2).*v2(:,1);
      % seamanj added: dbl for double
  elseif size(V,2) == 3
      %n = cross(v1,v2,2); dblA = multinorm(n,2);

      % area of parallelogram is twice area of triangle
      % area of parallelogram is || v1 x v2 || 
      n  = cross(v1,v2,2); 
      % THIS DOES MATRIX NORM!!! don't use it!!
      % 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
  % cotangents and diagonal entries for element matrices
  cot12 = -dot(v1,v2,2)./dblA/2; cot23 = -dot(v2,v3,2)./dblA/2; cot31 = -dot(v3,v1,2)./dblA/2;
  % diag entries computed from the condition that rows of the matrix sum up
  % to 0
  % (follows from the element matrix formula E_{ij} = (v_i dot v_j)/4/A )
  diag1 = -cot12-cot31; diag2 = -cot12-cot23; diag3 = -cot31-cot23;
  % indices of nonzero elements in the matrix for sparse() constructor
  i = [i1 i2 i2 i3 i3 i1 i1 i2 i3];
  j = [i2 i1 i3 i2 i1 i3 i1 i2 i3];
  % values corresponding to pairs form (i,j)
  v = [cot12 cot12 cot23 cot23 cot31 cot31 diag1 diag2 diag3];
  % for repeated indices (i,j) sparse automatically sums up elements, as we
  % want
  L = sparse(i,j,v,size(V,1),size(V,1));
end

I have already added some comments in the code. Therefore, I would like to pick some important lines to explain.

For one edge in each triangle, say edge e12 , the corresponding element in stiffness matrix L12 is like:

L12=∫Ω−∇ϕ1∇ϕ2dA