zzh1161 / linearhydroana

Linear hydrostatic finite element analysis: tetrahedral and hexahedral simulation.

Linear Hydrostatic Analysis

File Description

• include contains all the header files.

  • info.hpp contains some macro definitions.
  • LinSysSolver.hpp defines an abstract base class that stores sparse matrices in CSR format for the assembly of stiffness matrices and the solution of linear systems.
    EigenLibSolver.hpp and CHOLMODSolver.hpp are the corresponding inherited classes, representing the use of the Eigen library or CHOLMOD library to solve.
  • readTetMesh.hpp defines the function to read the tetrahedral mesh.msh file.
    readHexMesh.hpp defines the function to read the hexahedral mesh .in files, where the .in files are parsed from the .vtk files.
    readCondition.hpp defines function to read boundary conditions and other information, with requirements for the input file format.
  • shapeInterface.hpp defines the abstract base class for the grid.
    tet4nodeEle.hpp is an inherited 4-node tetrahedron class.
    hex8nodeEle.hpp is an inherited 8-node hexahedron class.
  • tet4nodeSolver.hpp encapsulates the finite element analysis process of a tetrahedral mesh model.
    hex8nodeSolver.hpp encapsulates the finite element analysis process of a hexahedral mesh model.

• tests contains all test examples.

  • tet_cube tests a simple mesh model with 8 vertices and 8 tetrahedra, and the correctness was verified by a matlab program.
  • tet_bunny tests the tetrahedral mesh model bunny.msh.
  • tet_armadillo tests the tetrahedral mesh model Armadillo219K.msh, which has 219k nodes and takes about 30 minutes to solve using the Eigen library and only 5 minutes using the CHOLMOD library.
  • hex_smallCube tests a small cube model consisting of 27 small hexahedra and verified the correctness using matlab.
  • hex_cube tests a larger cube model consisting of 512 hexahedra with a total of 729 nodes.
  • hex_bunny7k tests a bunny model consisting of 7k hexahedra with a total of 9k nodes.

Compile and Run

The code author has the following environment and software locally:

• Ubuntu 20.04
• g++(GCC) 11.2.0 (C++20)
• Python 3.8.10
• CMake 3.16.3
• Eigen 3.3.7
• SuiteSparse 5.7.1
• MshIO
• meshio 5.3.4

Compiling:

• In the hex test samples, type the command

  mkdir build && cd build
  cmake ..
  make

  to compile. Type ./main to run it.

• In the tet samples, type make to compile. Type ./main to run it.

Theoretical Analysis

Spatial 4-node Tetrahedral Unit

1. node description

   • geometric coordinates: $(x_i, y_i, z_i)$
   • node displacement vector: $[u_1, v_1, w_1, u_2, v_2, w_2, u_3, v_3, w_3, u_4, v_4, w_4]^T$
   • node force vector: $[P_{x1}, P_{y1}, P_{z1}, P_{x2}, P_{y2}, P_{z2}, P_{x3}, P_{y3}, P_{z3}, P_{x4}, P_{y4}, P_{z4}]^T$

2. displacement field description

   $$ \begin{align*} u(x,y,z) &= a_0+a_1x+a_2y+a_3z \ v(x,y,z) &= b_0+b_1x+b_2y+b_3z \ w(x,y,z) &= c_0+c_1x+c_2y+c_3z \end{align*} $$

   $$ \begin{align*} u(x,y,z) &= N_1(x,y,z)u_1+N_2u_2+N_3u_3+N_4u_4 \ v(x,y,z) &= N_1(x,y,z)v_1+N_2v_2+N_3v_3+N_4v_4 \ w(x,y,z) &= N_1(x,y,z)w_1+N_2w_2+N_3w_3+N_4w_4 \end{align*} $$

   $$ \begin{align*} \textbf{u}(x,y,z) &= \begin{bmatrix} u(x,y,z) \ v(x,y,z) \ w(x,y,z) \end{bmatrix} = \textbf{N}(x,y,z)\cdot \textbf{q}^e \ &= \begin{bmatrix} N_1 & 0 & 0 & N_2 & 0 & 0 & N_3 & 0 & 0 & N_4 & 0 & 0 \ 0 & N_1 & 0 & 0 & N_2 & 0 & 0 & N_3 & 0 & 0 & N_4 & 0 \ 0 & 0 & N_1 & 0 & 0 & N_2 & 0 & 0 & N_3 & 0 & 0 & N_4 \end{bmatrix}\cdot \textbf{q}^e \end{align*} $$

3. strain field

   $$ \begin{align*} \mathbf{\epsilon}(x,y,z) = \begin{bmatrix} \epsilon_{xx} \ \epsilon_{yy} \ \epsilon_{zz} \ \epsilon_{xy} \ \epsilon_{yz} \ \epsilon_{zx} \end{bmatrix} = \begin{bmatrix} \frac{\partial}{\partial x} & 0 & 0\ 0 & \frac{\partial}{\partial y} & 0\ 0 & 0 & \frac{\partial}{\partial z}\ \frac{\partial}{\partial y} & \frac{\partial}{\partial x} & 0\ 0 & \frac{\partial}{\partial z} & \frac{\partial}{\partial y}\ \frac{\partial}{\partial z} & 0 & \frac{\partial}{\partial x} \end{bmatrix}\begin{bmatrix}u\ v \ w\end{bmatrix} = [\partial]\mathbf{N}\cdot\mathbf{q}^e = \mathbf{B}\cdot\mathbf{q}^e \end{align*} $$

4. stress field

   $$ \mathbf{\sigma} = \mathbf{D}\cdot\mathbf{\epsilon} = \mathbf{D}\cdot\mathbf{B}\cdot\mathbf{q}^e=\mathbf{S}\cdot\mathbf{q}^e $$

5. principle of minimum potential energy

   $$ \begin{align*} \Pi^e &= \frac{1}{2}\int_{\Omega^e}\mathbf{\sigma}^T\mathbf{\epsilon}\ d\Omega - \left[\int_{\Omega^e}\overline{\mathbf{b}}^T\mathbf{u}\ d\Omega+\int_{S^e_p}\overline{\mathbf{p}}^T\mathbf{u}\ dA\right] \ &= \frac{1}{2}\mathbf{q}^{eT}\left(\int_{\Omega^e}\mathbf{B}^T\mathbf{DB}\ d\Omega\right)\mathbf{q}^e - \left(\int_{\Omega^e}\mathbf{N}^T\overline{\mathbf{b}}\ d\Omega+\int_{S^e_q}\mathbf{N}^T\overline{\mathbf{p}}\ dA\right)^T\mathbf{q}^e \ &= \frac{1}{2}\mathbf{q}^{eT}\mathbf{K}^e\mathbf{q}^e - \mathbf{P}^{eT}\mathbf{q}^e \end{align*} $$

Spatial 8-node Hexahedron Unit

The analysis process is similar to the above process.

Simulation Results

1. tet_bunny

   

   This is a comparison of the results obtained by fixing the bottom of the rabbit model and applying a horizontal force on its ears.

2. tet_armadillo

   

   This is a comparison of the results obtained by fixing the lower limbs of the model and applying a horizontal inward force on its hands.

3. hex_bunny7k

   

   This is a comparison of the results obtained by fixing the bottom of the rabbit model(with about 7k nodes) and applying a horizontal force on its ears.

4. hex_bunny10w

   

   This is a comparison of the results obtained by fixing the bottom of the rabbit model(with about 110k nodes) and applying a horizontal force on its ears.