Problems

• global stiffness K is neither diagonally dominant nor has a spectral radius of <1 ...
• how to do gaussian quadrature for matrix integrand?
• how to correctly enforce dirichlet boundary
  • set large value to K_ii for vertex i on boundary
  • set K_ii to 1 and b_i to 0 for i on boundary, also removes entries K_{ij},K_{ji} for all j
  • lagrange multiplier
  • projection matrix to contain only unconstrained unknowns
• convergence speed is slow for classical iterative methods
  • slow for large A
    • e^k <= \rho(A) ~ 1 - c/d^2 (nearly 1 for large d) for some constant c and d is dimension of matrix A
• if cannot get preconditioning to work, work on small

what kind of experiments

• linear solvers
  • backslash (ground truth)
  • gauss-seidel

Meetings

Questions

• what are basis function look like in 3d
  • simply 1d hat function in each spatial coordinate ?
  • complications with using voxels (i.e. box/hexahedral)
    • virtual element methods ?
• fem enforces natural boundary,
  • imagine 3d mesh, want to find displacement field
  • u fixed on feet, but free otherwise
  • however fdm requires boundary known for the entire boundary, which seem to be impossible to do with fdm?
  • answer: always natural boundary with displacement at nodes not touching ground
    • one more unknown with one more equation
    • taylor approximation with 3 poitns near boundary to approximate differnetail
• what are useful iterative methods (jacobi, conjugate gradient) to use
  • consider parallelism
  • where to look for resources
• alternative idea
  • if 3d basis for polyhedra is hard to implement,
    • why not just destruct each voxel into 6 tetrahedra, the local stiffness matrix would be same,
    • simply need to compute local stiffness matrix 6 times, and populate sparse rhs
    • otherwise, matrix-free iterative method
• time-wise possible?
  • 3d impl of fem?
  • multigrid?
  • gpu implementation?
  • just pick one!
• boundary of voxel, model changes ?
  • need more research on solving pde on voxels
  • what happens at boundary

Meetings

• fast real=time
  • reconstruct part of the mesh that changes during deformation
  • initial guess to linear elasticity
• linear solver
  • preconditioning: reduce number of iteration for conjugate gradient
    • incomplete cholesky for conjugate gradient solver
  • multigrid methods
    • O(N)
    • might be hard to implement ourselves
• hex
  • mesh refinement in areas where error is large
    • monitor function, hints to where place are hard
    • i.e. max_[x_k,x_{k+1}] |x(x)''|
    • adaptive method for PDE ...
  • adaptive mesh refinement to reduce error ...
    • i.e. prespecify the grid, or dynamically adjust things itself
• our idea
  • solution to previous problem
    • recomputing parts of A
    • supply solution from previous iteration as starting guess ...
  • a better iterative method
    • conjugate gradient
    • multigrid
• 446 project
  • error analysis
• how quickly conjugate gradient converge
  • depends on condition number kappa
  • iteration: sqrt(kappa(A))
  • with preconditioner: sqrt(kappa(M-1 A))
  • monitor residule Ae = r, computable with r=Ax-b
  • need to find softwares ...

Junk

% per element stress field
se = zeros(12,1);
straine = zeros(size(Tet,1),6);
NV = zeros(size(V,1),1);    % #Vx1 volume of sum of neighboring tets

for i = 1:size(Tet,1)
    B = squeeze(Bs(i,:,:));
    Teti = Tet(i,:);
    ij2p(1:3:end) = 3*Teti-2;
    ij2p(2:3:end) = 3*Teti-1;
    ij2p(3:3:end) = 3*Teti;
    straine(i,:) = B*u(ij2p);
    NV(Teti) = NV(Teti) + Vs(i);
end

% per vertex strain/stress
strain = zeros(size(V,1),6);
stress = zeros(size(V,1),6);

for i =1:size(Tet,1)
    Teti = Tet(i,:);
    strain(Teti,:) = strain(Teti,:) + (1./NV(Teti))*(Vs(i).*straine(i,:));
end

for j = 1:size(V,1)
    stress(j,:) = (C*strain(j,:)')';
end

% von mises
se = zeros(6,1);
VM = zeros(size(V,1),1);
for j = 1:size(V,1)
    se = stress(j,:);
    VM(j) = (1./sqrt(2))*sqrt((se(1)-se(2))^2 + (se(2)-se(3))^2 + ...
        (se(3)-se(1))^2 + 6*(se(4)^2+se(5)^2+se(6)^2));
end

U large and challenge to learn PCA v.s. Autoencoder L-BFGS linear model reduction: nonlinear?

Diffusion curves

Diffusion of colors smooth gradient = minimize surface curvature