4b665fa82b
URDF import demo: add COLLADA .dae file support add FiniteElementMethod demo, extracted from the OpenTissue library (under the zlib license) don't crash if loading an invalid STL file add comparison with Assimp for COLLADA file loading (disabled by default, to avoid library dependency) Gwen: disable some flags that make the build incompatible
176 lines
7.7 KiB
C++
176 lines
7.7 KiB
C++
#ifndef OPENTISSUE_DYNAMICS_FEM_FEM_COMPUTE_MASS_H
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#define OPENTISSUE_DYNAMICS_FEM_FEM_COMPUTE_MASS_H
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//
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// OpenTissue Template Library
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// - A generic toolbox for physics-based modeling and simulation.
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// Copyright (C) 2008 Department of Computer Science, University of Copenhagen.
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//
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// OTTL is licensed under zlib: http://opensource.org/licenses/zlib-license.php
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//
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#include <OpenTissue/configuration.h>
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#include <limits>
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namespace OpenTissue
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{
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namespace fem
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{
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namespace detail
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{
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/**
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* Compute Mass.
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*
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* Note: Volume should have been computed prior to invoking this
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* function. Thus make sure that initialize_stiffness_elements have
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* been invoked prior to this function.
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*
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* @param mesh
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*/
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template<typename fem_mesh>
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inline void compute_mass(fem_mesh & mesh)
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{
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typedef typename fem_mesh::real_type real_type;
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//---
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//--- A mass matrix is a discrete representation of a continuous mass distribution.
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//--- To compute our mass matrix for a tetrahedral element with linear shape
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//--- functions we need the formula (pp. 266 in Cook)
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//---
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//--- a!b!c!d!
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//--- \int_V N_1^a N_2^b N_3^c N_4^d dV = 6V -------------------------- (**)
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//--- (3 + a + b +c + d)!
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//---
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//--- A consistent element mass matrix (pp. 376 Cook) is defined as
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//---
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//--- m = \int_V \rho N^T N dV (***)
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//---
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//--- This equation can be derived from work balance, the details of which is unimportant
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//--- here (look Cook pp. 375-376 for details).
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//--- Assumping \rho is constant over each tetrahedral element and using the linear shape
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//--- functions the above definition (***) results in
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//---
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//--- |N_1|
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//--- m = \rho \int_V |N_2| |N_1 N_2 N_3 N_4| dV
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//--- |N_3|
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//--- |N_4|
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//---
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//--- |(N_1 N_1) (N_1 N_2) (N_1 N_3) (N_1 N_4)|
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//--- m = \rho \int_V |(N_2 N_1) (N_2 N_2) (N_2 N_3) (N_2 N_4)| dV
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//--- |(N_3 N_1) (N_3 N_2) (N_3 N_3) (N_3 N_4)|
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//--- |(N_4 N_1) (N_4 N_2) (N_4 N_3) (N_4 N_4)|
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//---
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//--- by (**)
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//---
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//--- | 2 1 1 1|
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//--- m = \rho V/20 | 1 2 1 1| (****)
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//--- | 1 1 2 1|
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//--- | 1 1 1 2|
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//---
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//--- V
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//--- m_ij = \rho --- (1+delta_ij)
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//--- 20
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//---
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//--- in 3D this means that for the tetrahedral element
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//---
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//--- | 2 2 2 1 1 1 1 1 1 1 1 1 |
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//--- | 2 2 2 1 1 1 1 1 1 1 1 1 |
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//--- | 2 2 2 1 1 1 1 1 1 1 1 1 |
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//--- | |
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//--- | 1 1 1 2 2 2 1 1 1 1 1 1 |
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//--- | 1 1 1 2 2 2 1 1 1 1 1 1 |
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//--- V | 1 1 1 2 2 2 1 1 1 1 1 1 |
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//--- Me = \rho --- | |
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//--- 20 | 1 1 1 1 1 1 2 2 2 1 1 1 |
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//--- | 1 1 1 1 1 1 2 2 2 1 1 1 |
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//--- | 1 1 1 1 1 1 2 2 2 1 1 1 |
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//--- | |
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//--- | 1 1 1 1 1 1 1 1 1 2 2 2 |
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//--- | 1 1 1 1 1 1 1 1 1 2 2 2 |
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//--- | 1 1 1 1 1 1 1 1 1 2 2 2 |
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//---
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//--- Notice that in order to obtain the global/system mass matrix an assembly similar to the
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//--- stiffnees matrix assembly must be carried out. Further, the global M matrix will
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//--- have the same sub-block pattern as the global K matrix.
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//---
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//--- A consistent mass matrix is often not used in computer graphics. Instead and
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//--- ad-hoc approach named ``lumped'' mass matrix is applied.
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//--- The lumped mass matrix is obtained by placing particle masses at the nodes.
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//--- This corresponds to shifting all the masses in the rows of (****) onto the
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//--- diagonal. In 3D this yields the element mass matrix
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//---
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//--- | 1 0 0 0 0 0 0 0 0 0 0 0 |
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//--- | 0 1 0 0 0 0 0 0 0 0 0 0 |
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//--- | 0 0 1 0 0 0 0 0 0 0 0 0 |
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//--- | |
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//--- | 0 0 0 1 0 0 0 0 0 0 0 0 |
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//--- | 0 0 0 0 1 0 0 0 0 0 0 0 |
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//--- V | 0 0 0 0 0 1 0 0 0 0 0 0 |
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//--- Me = \rho --- | |
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//--- 4 | 0 0 0 0 0 0 1 0 0 0 0 0 |
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//--- | 0 0 0 0 0 0 0 1 0 0 0 0 |
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//--- | 0 0 0 0 0 0 0 0 1 0 0 0 |
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//--- | |
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//--- | 0 0 0 0 0 0 0 0 0 1 0 0 |
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//--- | 0 0 0 0 0 0 0 0 0 0 1 0 |
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//--- | 0 0 0 0 0 0 0 0 0 0 0 1 |
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//---
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//--- Thus a lumped mass matrix is diagonal whereas a consistent mass matrix
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//--- is not. Observe that the global mass matrix would also diagonal and the
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//--- assembly is simplified to an iteration over all tetrahedra, while
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//--- incementing the nodal mass by one fourth of the tetrahedral mass.
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//---
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//--- for each node n
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//--- mass(n) = 0
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//--- next n
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//--- for each tetrahedron e
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//--- mass(n_i) += \rho_e Ve / 4
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//--- mass(n_j) += \rho_e Ve / 4
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//--- mass(n_k) += \rho_e Ve / 4
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//--- mass(n_m) += \rho_e Ve / 4
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//--- next e
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//---
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//--- where n_i,n_j,n_k and n_m are the four nodes of the e'th tetrahedron.
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//---
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//--- The advantage of lumping is less storage and higher performace. On the downside
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//--- lumping introduces a discontinouty in the displacement field.
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//---
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//--- Obrien.shen state that the errors in lumping is negligeble for small-size course
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//--- meshes used in computer graphics. However, for finer meshes the errors becomes
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//--- noticeable.
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//---
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//--- There do exist other approaches for computing mass matrices, even mehtods which
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//--- combine other methods. We refer the interested reader to Cook for more details. Here
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//--- we have limited our selfes to the two most common methods.
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//---
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//--- It is worthwhile to notice that under the reasonable assumptions that V and \rho are
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//--- positive for all elements both the element mass matrices and the global mass matrices
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//--- are symmetric positive definite matrices.
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for (int n=0;n<mesh.m_nodes.size();n++)
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{
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if(mesh.m_nodes[n].m_fixed)
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mesh.m_nodes[n].m_mass = std::numeric_limits<real_type>::max();
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else
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mesh.m_nodes[n].m_mass = 0;
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}
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for (int t=0;t<mesh.m_tetrahedra.size();t++)
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{
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real_type amount = mesh.m_tetrahedra[t].m_density * mesh.m_tetrahedra[t].m_V *.25;
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mesh.m_tetrahedra[t].i()->m_mass += amount;
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mesh.m_tetrahedra[t].j()->m_mass += amount;
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mesh.m_tetrahedra[t].k()->m_mass += amount;
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mesh.m_tetrahedra[t].m()->m_mass += amount;
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}
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}
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} // namespace detail
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} // namespace fem
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} // namespace OpenTissue
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//OPENTISSUE_DYNAMICS_FEM_FEM_COMPUTE_MASS_H
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#endif
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