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fem.c
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fem.c
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/*
* fem.c
* Copyright (C) 2008, Tomasz Koziara (t.koziara AT gmail.com)
* --------------------------------------------------------------
* finite element method
*/
/* This file is part of Solfec.
* Solfec is free software: you can redistribute it and/or modify it under
* the terms of the GNU Lesser General Public License as published by the
* Free Software Foundation, either version 3 of the License, or (at your
* option) any later version.
*
* Solfec is distributed in the hope that it will be useful, but WITHOUT
* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
* FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
* License for more details.
*
* You should have received a copy of the GNU Lesser General Public
* License along with Solfec. If not, see <http://www.gnu.org/licenses/>. */
#include <string.h>
#include <float.h>
#include "lap.h"
#include "mem.h"
#include "sol.h"
#include "fem.h"
#include "alg.h"
#include "bla.h"
#include "hyb.h"
#include "cvi.h"
#include "gjk.h"
#include "kdt.h"
#include "svk.h"
#include "but.h"
#include "err.h"
#if OMP
#include <omp.h>
#include "ompu.h"
#endif
typedef double (*node_t) [3]; /* mesh node */
#define IMP_EPS 1E-9
#define MAX_ITERS 64
#define MAX_NODES 20
#define DOM_TOL 0.1
#define CUT_TOL 0.001
#define MESH_DOFS(msh) ((msh)->nodes_count * 3)
#define FEM_MESH_CONF(bod) ((bod)->form < BODY_COROTATIONAL_MODAL ? (bod)->conf : (bod)->conf + (bod)->dofs + 9) /* mesh space configuration */
#define FEM_MESH_VELO(bod) ((bod)->form < BODY_COROTATIONAL_MODAL ? (bod)->velo : (bod)->velo + (bod)->dofs * 4) /* mesh space velocity */
#define FEM_MESH_VEL0(bod) (FEM_MESH_VELO(bod) + MESH_DOFS(FEM_MESH(bod))) /* mesh space previous velocity */
#define FEM_MESH_MASS(bod) ((bod)->form < BODY_COROTATIONAL_MODAL ? (bod)->M->x : ((bod)->conf + (bod)->dofs + 9 + MESH_DOFS(FEM_MESH(bod)))) /* mesh space velocity */
#define FEM_VEL0(bod) ((bod)->velo + (bod)->dofs) /* previous velocity */
#define FEM_FEXT(bod) ((bod)->velo + (bod)->dofs * 2) /* external force */
#define FEM_FINT(bod) ((bod)->velo + (bod)->dofs * 3) /* internal force */
#define FEM_FBOD(bod) ((bod)->form < BODY_COROTATIONAL_MODAL ? (bod)->velo + (bod)->dofs * 4 : FEM_MESH_VEL0(bod) + MESH_DOFS(FEM_MESH(bod))) /* unit body force */
#define FEM_ROT(bod) ((bod)->conf + (bod)->dofs) /* rotation */
/* ==================== INTEGRATION ======================= */
/* order 1 */
static const double I_TET1_X[] = {0.25};
static const double I_TET1_Y[] = {0.25};
static const double I_TET1_Z[] = {0.25};
static const double I_TET1_W[] = {0.16666666666666666};
#define I_TET1_N 1
static const double I_PYR1_X[] = {0.00};
static const double I_PYR1_Y[] = {0.00};
static const double I_PYR1_Z[] = {0.25};
static const double I_PYR1_W[] = {1.333333333333333333};
#define I_PYR1_N 1
static const double I_WED1_X[] = {0.25};
static const double I_WED1_Y[] = {0.25};
static const double I_WED1_Z[] = {0.00};
static const double I_WED1_W[] = {0.333333333333333333};
#define I_WED1_N 1
static const double I_HEX1_X[] = {0.0};
static const double I_HEX1_Y[] = {0.0};
static const double I_HEX1_Z[] = {0.0};
static const double I_HEX1_W[] = {8.0};
#define I_HEX1_N 1
static const double I_TRI1_X[] = {0.333333333333333333};
static const double I_TRI1_Y[] = {0.333333333333333333};
static const double I_TRI1_W[] = {0.5};
#define I_TRI1_N 1
static const double I_QUA1_X [] = {0.0};
static const double I_QUA1_Y [] = {0.0};
static const double I_QUA1_W [] = {4.0};
#define I_QUA1_N 1
/* order 2 */
static const double I_TET2_X [] = {0.13819660112501052, 0.13819660112501052, 0.13819660112501052, 0.58541019662496840};
static const double I_TET2_Y [] = {0.13819660112501052, 0.13819660112501052, 0.58541019662496840, 0.13819660112501052};
static const double I_TET2_Z [] = {0.13819660112501052, 0.58541019662496840, 0.13819660112501052, 0.13819660112501052};
static const double I_TET2_W [] = {0.04166666666666666, 0.04166666666666666, 0.04166666666666666, 0.04166666666666666};
#define I_TET2_N 4
#define ISQR3_PYR1 0.263184055569714
#define ISQR3_PYR2 0.506616303349788
#define _1SUB_PYR1 0.544151844011225
#define _1SUB_PYR2 0.122514822655441
#define _1_WPYR1 0.100785882079825
#define _1_WPYR2 0.232547451253508
static const double I_PYR2_X[] = {-ISQR3_PYR1, ISQR3_PYR1, ISQR3_PYR1, -ISQR3_PYR1, -ISQR3_PYR2, ISQR3_PYR2, ISQR3_PYR2, -ISQR3_PYR2};
static const double I_PYR2_Y[] = {-ISQR3_PYR1, -ISQR3_PYR1, ISQR3_PYR1, ISQR3_PYR1, -ISQR3_PYR2, -ISQR3_PYR2, ISQR3_PYR2, ISQR3_PYR2};
static const double I_PYR2_Z[] = {_1SUB_PYR1, _1SUB_PYR1, _1SUB_PYR1, _1SUB_PYR1, _1SUB_PYR2, _1SUB_PYR2, _1SUB_PYR2, _1SUB_PYR2};
static const double I_PYR2_W[] = {_1_WPYR1, _1_WPYR1, _1_WPYR1, _1_WPYR1, _1_WPYR2, _1_WPYR2, _1_WPYR2, _1_WPYR2};
#define I_PYR2_N 8
#define ISQR3 0.57735026918962584
static const double I_WED2_X[] = {0.66666666666666667, 0.16666666666666667, 0.16666666666666667, 0.66666666666666667, 0.16666666666666667, 0.16666666666666667};
static const double I_WED2_Y[] = {0.16666666666666667, 0.66666666666666667, 0.16666666666666667, 0.16666666666666667, 0.66666666666666667, 0.16666666666666667};
static const double I_WED2_Z[] = {-ISQR3, -ISQR3, -ISQR3, ISQR3, ISQR3, ISQR3};
static const double I_WED2_W[] = {0.16666666666666667, 0.16666666666666667, 0.16666666666666667, 0.16666666666666667, 0.16666666666666667, 0.16666666666666667};
#define I_WED2_N 6
static const double I_HEX2_X [] = {-ISQR3, ISQR3, ISQR3, -ISQR3, -ISQR3, ISQR3, ISQR3, -ISQR3};
static const double I_HEX2_Y [] = {-ISQR3, -ISQR3, ISQR3, ISQR3, -ISQR3, -ISQR3, ISQR3, ISQR3};
static const double I_HEX2_Z [] = {-ISQR3, -ISQR3, -ISQR3, -ISQR3, ISQR3, ISQR3, ISQR3, ISQR3};
static const double I_HEX2_W [] = {1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0};
#define I_HEX2_N 8
static const double I_TRI2_X[] = {0.66666666666666667, 0.16666666666666667, 0.16666666666666667};
static const double I_TRI2_Y[] = {0.16666666666666667, 0.66666666666666667, 0.16666666666666667};
static const double I_TRI2_W[] = {0.16666666666666667, 0.16666666666666667, 0.16666666666666667};
#define I_TRI2_N 3
static const double I_QUA2_X [] = {-ISQR3, ISQR3, ISQR3, -ISQR3};
static const double I_QUA2_Y [] = {-ISQR3, -ISQR3, ISQR3, ISQR3};
static const double I_QUA2_W [] = {1.0, 1.0, 1.0, 1.0};
#define I_QUA2_N 4
/* order 3 */
static const double I_HEX3_X [] = {0.7745966692414834, -0.7745966692414834, -0.7745966692414834, -0.7745966692414834, -0.7745966692414834, -0.7745966692414834, -0.7745966692414834, -0.7745966692414834, -0.7745966692414834, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.7745966692414834, 0.7745966692414834, 0.7745966692414834, 0.7745966692414834, 0.7745966692414834, 0.7745966692414834, 0.7745966692414834, 0.7745966692414834, 0.7745966692414834};
static const double I_HEX3_Y [] = {-0.7745966692414834, -0.7745966692414834, -0.7745966692414834, 0.0, 0.0, 0.0, 0.7745966692414834, 0.7745966692414834, 0.7745966692414834, -0.7745966692414834, -0.7745966692414834, -0.7745966692414834, 0.0, 0.0, 0.0, 0.7745966692414834, 0.7745966692414834, 0.7745966692414834, -0.7745966692414834, -0.7745966692414834, -0.7745966692414834, 0.0, 0.0, 0.0, 0.7745966692414834, 0.7745966692414834, 0.7745966692414834};
static const double I_HEX3_Z [] = {-0.7745966692414834, 0.0, 0.7745966692414834, -0.7745966692414834, 0.0, 0.7745966692414834, -0.7745966692414834, 0.0, 0.7745966692414834, -0.7745966692414834, 0.0, 0.7745966692414834, -0.7745966692414834, 0.0, 0.7745966692414834, -0.7745966692414834, 0.0, 0.7745966692414834, -0.7745966692414834, 0.0, 0.7745966692414834, -0.7745966692414834, 0.0, 0.7745966692414834, -0.7745966692414834, 0.0, 0.7745966692414834};
static const double I_HEX3_W [] = {0.1714677640603567, 0.27434842249657065, 0.1714677640603567, 0.27434842249657065, 0.43895747599451296, 0.27434842249657065, 0.1714677640603567, 0.27434842249657065, 0.1714677640603567, 0.27434842249657065, 0.43895747599451296, 0.27434842249657065, 0.438957475994513, 0.7023319615912208, 0.438957475994513, 0.27434842249657065, 0.43895747599451296, 0.27434842249657065, 0.1714677640603567, 0.27434842249657065, 0.1714677640603567, 0.27434842249657065, 0.43895747599451296, 0.27434842249657065, 0.1714677640603567, 0.27434842249657065, 0.1714677640603567};
#define I_HEX3_N 27
/* order 4 */
static const double I_HEX4_X [] = {-0.8611363115940526, -0.8611363115940526, -0.8611363115940526, -0.8611363115940526, -0.8611363115940526, -0.8611363115940526, -0.8611363115940526, -0.8611363115940526, -0.8611363115940526, -0.8611363115940526, -0.8611363115940526, -0.8611363115940526, -0.8611363115940526, -0.8611363115940526, -0.8611363115940526, -0.8611363115940526, -0.3399810435848563, -0.3399810435848563, -0.3399810435848563, -0.3399810435848563, -0.3399810435848563, -0.3399810435848563, -0.3399810435848563, -0.3399810435848563, -0.3399810435848563, -0.3399810435848563, -0.3399810435848563, -0.3399810435848563, -0.3399810435848563, -0.3399810435848563, -0.3399810435848563, -0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.8611363115940526, 0.8611363115940526, 0.8611363115940526, 0.8611363115940526, 0.8611363115940526, 0.8611363115940526, 0.8611363115940526, 0.8611363115940526, 0.8611363115940526, 0.8611363115940526, 0.8611363115940526, 0.8611363115940526, 0.8611363115940526, 0.8611363115940526, 0.8611363115940526, 0.8611363115940526};
static const double I_HEX4_Y [] = {-0.8611363115940526, -0.8611363115940526, -0.8611363115940526, -0.8611363115940526, -0.3399810435848563, -0.3399810435848563, -0.3399810435848563, -0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.8611363115940526, 0.8611363115940526, 0.8611363115940526, 0.8611363115940526, -0.8611363115940526, -0.8611363115940526, -0.8611363115940526, -0.8611363115940526, -0.3399810435848563, -0.3399810435848563, -0.3399810435848563, -0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.8611363115940526, 0.8611363115940526, 0.8611363115940526, 0.8611363115940526, -0.8611363115940526, -0.8611363115940526, -0.8611363115940526, -0.8611363115940526, -0.3399810435848563, -0.3399810435848563, -0.3399810435848563, -0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.8611363115940526, 0.8611363115940526, 0.8611363115940526, 0.8611363115940526, -0.8611363115940526, -0.8611363115940526, -0.8611363115940526, -0.8611363115940526, -0.3399810435848563, -0.3399810435848563, -0.3399810435848563, -0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.3399810435848563, 0.8611363115940526, 0.8611363115940526, 0.8611363115940526, 0.8611363115940526};
static const double I_HEX4_Z [] = {-0.8611363115940526, -0.3399810435848563, 0.3399810435848563, 0.8611363115940526, -0.8611363115940526, -0.3399810435848563, 0.3399810435848563, 0.8611363115940526, -0.8611363115940526, -0.3399810435848563, 0.3399810435848563, 0.8611363115940526, -0.8611363115940526, -0.3399810435848563, 0.3399810435848563, 0.8611363115940526, -0.8611363115940526, -0.3399810435848563, 0.3399810435848563, 0.8611363115940526, -0.8611363115940526, -0.3399810435848563, 0.3399810435848563, 0.8611363115940526, -0.8611363115940526, -0.3399810435848563, 0.3399810435848563, 0.8611363115940526, -0.8611363115940526, -0.3399810435848563, 0.3399810435848563, 0.8611363115940526, -0.8611363115940526, -0.3399810435848563, 0.3399810435848563, 0.8611363115940526, -0.8611363115940526, -0.3399810435848563, 0.3399810435848563, 0.8611363115940526, -0.8611363115940526, -0.3399810435848563, 0.3399810435848563, 0.8611363115940526, -0.8611363115940526, -0.3399810435848563, 0.3399810435848563, 0.8611363115940526, -0.8611363115940526, -0.3399810435848563, 0.3399810435848563, 0.8611363115940526, -0.8611363115940526, -0.3399810435848563, 0.3399810435848563, 0.8611363115940526, -0.8611363115940526, -0.3399810435848563, 0.3399810435848563, 0.8611363115940526, -0.8611363115940526, -0.3399810435848563, 0.3399810435848563, 0.8611363115940526};
static const double I_HEX4_W [] = {0.04209147749053145, 0.07891151579507055, 0.07891151579507055, 0.04209147749053145, 0.07891151579507055, 0.1479403360567813, 0.1479403360567813, 0.07891151579507055, 0.07891151579507055, 0.1479403360567813, 0.1479403360567813, 0.07891151579507055, 0.04209147749053145, 0.07891151579507055, 0.07891151579507055, 0.04209147749053145, 0.07891151579507055, 0.1479403360567813, 0.1479403360567813, 0.07891151579507055, 0.1479403360567813, 0.27735296695391304, 0.27735296695391304, 0.1479403360567813, 0.1479403360567813, 0.27735296695391304, 0.27735296695391304, 0.1479403360567813, 0.07891151579507055, 0.1479403360567813, 0.1479403360567813, 0.07891151579507055, 0.07891151579507055, 0.1479403360567813, 0.1479403360567813, 0.07891151579507055, 0.1479403360567813, 0.27735296695391304, 0.27735296695391304, 0.1479403360567813, 0.1479403360567813, 0.27735296695391304, 0.27735296695391304, 0.1479403360567813, 0.07891151579507055, 0.1479403360567813, 0.1479403360567813, 0.07891151579507055, 0.04209147749053145, 0.07891151579507055, 0.07891151579507055, 0.04209147749053145, 0.07891151579507055, 0.1479403360567813, 0.1479403360567813, 0.07891151579507055, 0.07891151579507055, 0.1479403360567813, 0.1479403360567813, 0.07891151579507055, 0.04209147749053145, 0.07891151579507055, 0.07891151579507055, 0.04209147749053145};
#define I_HEX4_N 64
/* collective rules */
static const double *I_TET_X [] = {NULL, I_TET1_X, I_TET2_X};
static const double *I_TET_Y [] = {NULL, I_TET1_Y, I_TET2_Y};
static const double *I_TET_Z [] = {NULL, I_TET1_Z, I_TET2_Z};
static const double *I_TET_W [] = {NULL, I_TET1_W, I_TET2_W};
static const int I_TET_N [] = { 0, I_TET1_N, I_TET2_N};
static const double *I_PYR_X [] = {NULL, I_PYR1_X, I_PYR2_X};
static const double *I_PYR_Y [] = {NULL, I_PYR1_Y, I_PYR2_Y};
static const double *I_PYR_Z [] = {NULL, I_PYR1_Z, I_PYR2_Z};
static const double *I_PYR_W [] = {NULL, I_PYR1_W, I_PYR2_W};
static const int I_PYR_N [] = { 0, I_PYR1_N, I_PYR2_N};
static const double *I_WED_X [] = {NULL, I_WED1_X, I_WED2_X};
static const double *I_WED_Y [] = {NULL, I_WED1_Y, I_WED2_Y};
static const double *I_WED_Z [] = {NULL, I_WED1_Z, I_WED2_Z};
static const double *I_WED_W [] = {NULL, I_WED1_W, I_WED2_W};
static const int I_WED_N [] = { 0, I_WED1_N, I_WED2_N};
static const double *I_HEX_X [] = {NULL, I_HEX1_X, I_HEX2_X, I_HEX3_X, I_HEX4_X};
static const double *I_HEX_Y [] = {NULL, I_HEX1_Y, I_HEX2_Y, I_HEX3_Y, I_HEX4_Y};
static const double *I_HEX_Z [] = {NULL, I_HEX1_Z, I_HEX2_Z, I_HEX3_Z, I_HEX4_Z};
static const double *I_HEX_W [] = {NULL, I_HEX1_W, I_HEX2_W, I_HEX3_W, I_HEX4_W};
static const int I_HEX_N [] = { 0, I_HEX1_N, I_HEX2_N, I_HEX3_N, I_HEX4_N};
static const double *I_TRI_X [] = {NULL, I_TRI1_X, I_TRI2_X};
static const double *I_TRI_Y [] = {NULL, I_TRI1_Y, I_TRI2_Y};
static const double *I_TRI_W [] = {NULL, I_TRI1_W, I_TRI2_W};
static const int I_TRI_N [] = { 0, I_TRI1_N, I_TRI2_N};
static const double *I_QUA_X [] = {NULL, I_QUA1_X, I_QUA2_X};
static const double *I_QUA_Y [] = {NULL, I_QUA1_Y, I_QUA2_Y};
static const double *I_QUA_W [] = {NULL, I_QUA1_W, I_QUA2_W};
static const int I_QUA_N [] = { 0, I_QUA1_N, I_QUA2_N};
#define MAX_ORDER 4 /* integration order bound */
/* minimal local coords */
static double mincoord [9][3] =
{{0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0},
{0, 0, 0}, /* tet */
{-1, -1, 0}, /* pyr */
{0, 0, -1}, /* wed */
{0, 0, 0},
{-1, -1, -1}}; /* hex */
/* maximal local coords */
static double maxcoord [9][3] =
{{0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0},
{1, 1, 1}, /* tet */
{1, 1, 1}, /* pyr */
{1, 1, 1}, /* wed */
{0, 0, 0},
{1, 1, 1}}; /* hex */
/* load 3D integrator data */
inline static int integrator3d_load (int type, int order, const double **X, const double **Y, const double **Z, const double **W)
{
int N;
ASSERT_DEBUG (order >= 1 && order <= MAX_ORDER, "Integration order out of bounds");
switch (type)
{
case 4:
{
*X = I_TET_X [order];
*Y = I_TET_Y [order];
*Z = I_TET_Z [order];
*W = I_TET_W [order];
N = I_TET_N [order];
}
break;
case 5:
{
*X = I_PYR_X [order];
*Y = I_PYR_Y [order];
*Z = I_PYR_Z [order];
*W = I_PYR_W [order];
N = I_PYR_N [order];
}
break;
case 6:
{
*X = I_WED_X [order];
*Y = I_WED_Y [order];
*Z = I_WED_Z [order];
*W = I_WED_W [order];
N = I_WED_N [order];
}
break;
case 8:
{
*X = I_HEX_X [order];
*Y = I_HEX_Y [order];
*Z = I_HEX_Z [order];
*W = I_HEX_W [order];
N = I_HEX_N [order];
}
break;
}
return N;
}
/* predict 3D integration order */
typedef enum {MASS, BODF, INTF} ENTITY3D;
static int integrator3d_order (int type, ENTITY3D entity)
{
if (type == 8 && entity == MASS) return 4;
return 2; /* XXX */
}
/* load 2D integrator data */
inline static int integrator2d_load (int type, int order, const double **X, const double **Y, const double **W)
{
int N;
ASSERT_DEBUG (order >= 1 && order <= MAX_ORDER, "Integration order out of bounds");
switch (type)
{
case 3:
{
*X = I_TRI_X [order];
*Y = I_TRI_Y [order];
*W = I_TRI_W [order];
N = I_TRI_N [order];
}
break;
case 4:
{
*X = I_QUA_X [order];
*Y = I_QUA_Y [order];
*W = I_QUA_W [order];
N = I_QUA_N [order];
}
break;
}
return N;
}
/* predict 2D integration order */
static int integrator2d_order (int type)
{
if (type == 3) return 1;
else return 2; /* quads */
}
/* avoid warnings related to unsed point[], weight variables */
#if defined(__clang__)
#elif defined(__GNUC__) /* GCC */
#pragma GCC diagnostic ignored "-Wunused-but-set-variable"
#endif
/* element integration; note that below __t__->ver, __t__->center are in local element coordinates already */
#define INTEGRATE3D(TYPE, ENTITY, DOM, DOMNUM, ...)\
{\
const double *__X__, *__Y__, *__Z__, *__W__;\
double point [3], weight;\
int __N__, __k__, __l__;\
TRISURF *__d__ = DOM;\
\
if (__d__)\
{\
double __subnodes__ [4][3], __subJ__;\
double __subpoint__ [3];\
int __domnum__ = DOMNUM;\
TRI *__t__, *__e__;\
double __volume__;\
\
__N__ = integrator3d_load (4, 1, &__X__, &__Y__, &__Z__, &__W__);\
\
for (__volume__ = __k__ = 0; __k__ < __domnum__; __k__ ++) __volume__ += __d__ [__k__].volume;\
\
for (__l__ = 0; __l__ < __domnum__; __d__ ++, __l__ ++)\
{\
if (__d__->volume > DOM_TOL * __volume__)\
{\
COPY (__d__->center, __subnodes__ [3]);\
\
for (__t__ = __d__->tri, __e__ = __t__ + __d__->m; __t__ < __e__; __t__ ++)\
{\
COPY (__t__->ver [0], __subnodes__ [2]);\
COPY (__t__->ver [1], __subnodes__ [1]);\
COPY (__t__->ver [2], __subnodes__ [0]);\
\
for (__k__ = 0; __k__ < __N__; __k__ ++)\
{\
__subpoint__ [0] = __X__ [__k__];\
__subpoint__ [1] = __Y__ [__k__];\
__subpoint__ [2] = __Z__ [__k__];\
__subJ__ = element_det (4, __subnodes__, __subpoint__, NULL);\
tet_o1_local_to_global (__subnodes__, __subpoint__, point);\
weight = __subJ__ * __W__ [__k__];\
\
__VA_ARGS__\
}\
}\
}\
else\
{\
COPY (__d__->center, point);\
weight = __d__->volume;\
\
__VA_ARGS__\
}\
}\
}\
else\
{\
__N__ = integrator3d_load (TYPE, integrator3d_order (TYPE, ENTITY), &__X__, &__Y__, &__Z__, &__W__);\
for (__k__ = 0; __k__ < __N__; __k__ ++)\
{\
point [0] = __X__ [__k__];\
point [1] = __Y__ [__k__];\
point [2] = __Z__ [__k__];\
weight = __W__ [__k__];\
\
__VA_ARGS__\
}\
}\
}
/* face integral begins */
#define INTEGRAL2D_BEGIN(TYPE)\
{\
const double *__X__, *__Y__, *__W__;\
double point [2], weight;\
int __N__, __k__;\
\
__N__ = integrator2d_load (TYPE, integrator2d_order (TYPE), &__X__, &__Y__, &__W__);\
for (__k__ = 0; __k__ < __N__; __k__ ++)\
{\
point [0] = __X__ [__k__];\
point [1] = __Y__ [__k__];\
weight = __W__ [__k__];\
/* integration code in between
* uses point and weight values */
/* face integral ends */
#define INTEGRAL2D_END()\
}\
}
/* ==================== FACE ======================= */
/* face shape functions at a local 2-point */
inline static int face_shapes (FACE *fac, double *point, double *shapes)
{
switch (fac->type)
{
case 3: /* 1st order triangle */
shapes [0] = point [0];
shapes [1] = point [1];
shapes [2] = 1.0 - point [0] - point [1];
return 3;
case 4: /* 1st order quadrilateral */
shapes [0] = 0.25 * (1.0 - point [0]) * (1.0 - point [1]);
shapes [1] = 0.25 * (1.0 + point [0]) * (1.0 - point [1]);
shapes [2] = 0.25 * (1.0 + point [0]) * (1.0 + point [1]);
shapes [3] = 0.25 * (1.0 - point [0]) * (1.0 + point [1]);
return 4;
}
return 0;
}
/* face shape derivatices at a local 2-point */
inline static int face_derivs (FACE *fac, double *point, double *derivs)
{
switch (fac->type)
{
case 3:
derivs [0] = 1.0;
derivs [1] = 0.0;
derivs [2] = 0.0;
derivs [3] = 1.0;
derivs [4] = -1.0;
derivs [5] = -1.0;
return 3;
case 4:
derivs [0] = 0.25 * (point [1] - 1.0);
derivs [1] = 0.25 * (point [0] - 1.0);
derivs [2] = 0.25 * (1.0 - point [1]);
derivs [3] = -0.25 * (1.0 + point [0]);
derivs [4] = 0.25 * (1.0 + point [1]);
derivs [5] = 0.25 * (1.0 + point [0]);
derivs [6] = -0.25 * (1.0 + point [1]);
derivs [7] = 0.25 * (1.0 - point [0]);
return 4;
}
return 0;
}
/* compute face coordinates transformation determinant at a local 2-point */
inline static double face_det (FACE *fac, node_t nodes, double *point, double *normal)
{
double derivs [16], d0 [3], d1[3], tmp [3], *d;
int i, n;
n = face_derivs (fac, point, derivs);
if (!normal) normal = tmp;
SET (d0, 0);
SET (d1, 0);
for (i = 0, d = derivs; i < n; i ++, d += 2)
{
ADDMUL (d0, d[0], nodes[i], d0);
ADDMUL (d1, d[1], nodes[i], d1);
}
PRODUCT (d0, d1, normal);
return LEN (normal);
}
/* load face node coordinates into a local table */
inline static int element_nodes (node_t heap, int type, int *nodes, node_t stack);
#define face_nodes(heap, type, nodes, stack) element_nodes (heap, type, nodes, stack)
/* load face displacemnts into a local table */
inline static void face_displacements (double *heap, FACE *fac, double (*q) [3])
{
double *p;
int i;
for (i = 0; i < fac->type; i ++)
{
p = &heap [3 * fac->nodes [i]];
COPY (p, q[i]);
}
}
/* create face integration data;
* FACE->idata = {shapes0, normal0, J0*wgt0, shapes1, normal1, J1*wgt1, shapes2, normal2, J2*wgt2, shapes3, normal3, J3*wgt3} */
#define FACE_SHAPES(fac, i) ((fac)->idata+(i)*((fac)->type+4))
#define FACE_NORMAL(fac, i) (FACE_SHAPES(fac,i)+(fac)->type)
static void create_face_integration_data (MESH *msh)
{
double refn [4][3], J, *N, *shapes;
FACE *fac;
int n;
for (fac = msh->faces; fac; fac = fac->n)
{
n = 0; INTEGRAL2D_BEGIN (fac->type) { n ++; } INTEGRAL2D_END ()
ERRMEM (fac->idata = malloc (sizeof (double [n * (fac->type + 4)])));
face_nodes (msh->ref_nodes, fac->type, fac->nodes, refn);
INTEGRAL2D_BEGIN (fac->type) /* defines point and weight */
{
shapes = FACE_SHAPES (fac, __k__);
N = FACE_NORMAL (fac, __k__);
face_shapes (fac, point, shapes);
J = face_det (fac, refn, point, N);
N [3] = J * weight;
NORMALIZE (N);
}
INTEGRAL2D_END ()
}
}
/* =============================== ELEMENT ================================ */
/* linear tetrahedron shape functions */
inline static void tet_o1_shapes (double *point, double *shapes)
{
shapes [0] = 1.0 - (point [0] + point [1] + point [2]);
shapes [1] = point [0];
shapes [2] = point [1];
shapes [3] = point [2];
}
/* linear hexahedron shape functions */
inline static void hex_o1_shapes (double *point, double *shapes)
{
shapes [0] = 0.125 * (1.0 - point [0]) * (1.0 - point [1]) * (1.0 - point [2]);
shapes [1] = 0.125 * (1.0 + point [0]) * (1.0 - point [1]) * (1.0 - point [2]);
shapes [2] = 0.125 * (1.0 + point [0]) * (1.0 + point [1]) * (1.0 - point [2]);
shapes [3] = 0.125 * (1.0 - point [0]) * (1.0 + point [1]) * (1.0 - point [2]);
shapes [4] = 0.125 * (1.0 - point [0]) * (1.0 - point [1]) * (1.0 + point [2]);
shapes [5] = 0.125 * (1.0 + point [0]) * (1.0 - point [1]) * (1.0 + point [2]);
shapes [6] = 0.125 * (1.0 + point [0]) * (1.0 + point [1]) * (1.0 + point [2]);
shapes [7] = 0.125 * (1.0 - point [0]) * (1.0 + point [1]) * (1.0 + point [2]);
}
/* linear pyramid shape functions */
inline static void pyr_o1_shapes (double *point, double *shapes)
{
double ratio;
if (point [2] != 1.0) ratio = (point[0]*point[1]*point[2]) / (1 - point[2]);
else ratio = 0.0; /* XXX */
shapes [0] = 0.25 * ((1.0 + point[0])*(1.0 + point[1]) - point[2] + ratio); /* rabbit functions */
shapes [1] = 0.25 * ((1.0 - point[0])*(1.0 + point[1]) - point[2] - ratio);
shapes [2] = 0.25 * ((1.0 - point[0])*(1.0 - point[1]) - point[2] + ratio);
shapes [3] = 0.25 * ((1.0 + point[0])*(1.0 - point[1]) - point[2] - ratio);
shapes [4] = point [2];
}
/* linear wedge shape functions */
inline static void wed_o1_shapes (double *point, double *shapes)
{
shapes [0] = 0.5 * (1.0 - point[0] - point[1])*(1.0 - point[2]);
shapes [1] = 0.5 * point[0]*(1.0 - point[2]);
shapes [2] = 0.5 * point[1]*(1.0 - point[2]);
shapes [3] = 0.5 * (1.0 - point[0] - point[1])*(1.0 + point[2]);
shapes [4] = 0.5 * point[0]*(1.0 + point[2]);
shapes [5] = 0.5 * point[1]*(1.0 + point[2]);
}
/* linear tetrahedron shape derivatives */
inline static void tet_o1_derivs (double *point, double *derivs)
{
derivs [0] = -1.0;
derivs [1] = -1.0;
derivs [2] = -1.0;
derivs [3] = 1.0;
derivs [4] = 0.0;
derivs [5] = 0.0;
derivs [6] = 0.0;
derivs [7] = 1.0;
derivs [8] = 0.0;
derivs [9] = 0.0;
derivs [10] = 0.0;
derivs [11] = 1.0;
}
/* linear hexahedron shape derivatives */
inline static void hex_o1_derivs (double *point, double *derivs)
{
derivs[0] = -0.125 * (1 - point[1]) * (1 - point[2]);
derivs[1] = -0.125 * (1 - point[0]) * (1 - point[2]);
derivs[2] = -0.125 * (1 - point[0]) * (1 - point[1]);
derivs[3] = 0.125 * (1 - point[1]) * (1 - point[2]);
derivs[4] = -0.125 * (1 + point[0]) * (1 - point[2]);
derivs[5] = -0.125 * (1 + point[0]) * (1 - point[1]);
derivs[6] = 0.125 * (1 + point[1]) * (1 - point[2]);
derivs[7] = 0.125 * (1 + point[0]) * (1 - point[2]);
derivs[8] = -0.125 * (1 + point[0]) * (1 + point[1]);
derivs[9] = -0.125 * (1 + point[1]) * (1 - point[2]);
derivs[10] = 0.125 * (1 - point[0]) * (1 - point[2]);
derivs[11] = -0.125 * (1 - point[0]) * (1 + point[1]);
derivs[12] = -0.125 * (1 - point[1]) * (1 + point[2]);
derivs[13] = -0.125 * (1 - point[0]) * (1 + point[2]);
derivs[14] = 0.125 * (1 - point[0]) * (1 - point[1]);
derivs[15] = 0.125 * (1 - point[1]) * (1 + point[2]);
derivs[16] = -0.125 * (1 + point[0]) * (1 + point[2]);
derivs[17] = 0.125 * (1 + point[0]) * (1 - point[1]);
derivs[18] = 0.125 * (1 + point[1]) * (1 + point[2]);
derivs[19] = 0.125 * (1 + point[0]) * (1 + point[2]);
derivs[20] = 0.125 * (1 + point[0]) * (1 + point[1]);
derivs[21] = -0.125 * (1 + point[1]) * (1 + point[2]);
derivs[22] = 0.125 * (1 - point[0]) * (1 + point[2]);
derivs[23] = 0.125 * (1 - point[0]) * (1 + point[1]);
}
/* linear pyramid shape derivatives */
inline static void pyr_o1_derivs (double *point, double *derivs)
{
double drat0, drat1, drat2;
if (point [2] != 1.0)
{
drat0 = (point[1]*point[2]) / (1.0 - point[2]),
drat1 = (point[0]*point[2]) / (1.0 - point[2]),
drat2 = (point[0]*point[1]) / (1.0 - point[2])*(1.0 - point[2]);
}
else drat0 = drat1 = drat2 = 0.0; /* XXX */
derivs [0] = 0.25 * ((1.0 + point[1]) + drat0);
derivs [1] = 0.25 * ((1.0 + point[0]) + drat1);
derivs [2] = 0.25 * (-1.0 + drat2);
derivs [3] = 0.25 * (-(1.0 + point[1]) - drat0);
derivs [4] = 0.25 * ((1.0 - point[0]) - drat1);
derivs [5] = 0.25 * (-1.0 - drat2);
derivs [6] = 0.25 * (-(1.0 - point[1]) + drat0);
derivs [7] = 0.25 * (-(1.0 - point[0]) + drat1);
derivs [8] = 0.25 * (-1.0 + drat2);
derivs [ 9] = 0.25 * ((1.0 - point[1]) - drat0);
derivs [10] = 0.25 * (-(1.0 + point[0]) - drat1);
derivs [11] = 0.25 * (-1.0 - drat2);
derivs [12] = 0.0;
derivs [13] = 0.0;
derivs [14] = 1.0;
}
/* linear wedge shape derivatives */
inline static void wed_o1_derivs (double *point, double *derivs)
{
derivs [0] = -0.5 * (1.0 - point[2]);
derivs [1] = -0.5 * (1.0 - point[2]);
derivs [2] = -0.5 * (1.0 - point[0] - point[1]);
derivs [3] = 0.5 * (1.0 - point[2]);
derivs [4] = 0.0;
derivs [5] = -0.5 * point[0];
derivs [6] = 0.0;
derivs [7] = 0.5 * (1.0 - point[2]);
derivs [8] = -0.5 * point[1];
derivs [ 9] = -0.5 * (1.0 + point[2]);
derivs [10] = -0.5 * (1.0 + point[2]);
derivs [11] = 0.5 * (1.0 - point[0] - point[1]);
derivs [12] = 0.5 * (1.0 + point[2]);
derivs [13] = 0.0;
derivs [14] = 0.5 * point[0];
derivs [15] = 0.0;
derivs [16] = 0.5 * (1.0 + point[2]);
derivs [17] = 0.5 * point[1];
}
/* linear tetrahedron local to global point transformation */
static void tet_o1_local_to_global (node_t nodes, double *local, double *global)
{
double shapes [4];
int i, j;
tet_o1_shapes (local, shapes);
SET (global, 0.0);
for (i = 0; i < 3; i ++)
for (j = 0; j < 4; j ++)
global [i] += nodes [j][i] * shapes [j];
}
/* finite element shape functions; returns the number of nodes */
inline static int element_shapes (int type, double *point, double *shapes)
{
switch (type)
{
case 4: tet_o1_shapes (point, shapes); return 4;
case 5: pyr_o1_shapes (point, shapes); return 5;
case 6: wed_o1_shapes (point, shapes); return 6;
case 8: hex_o1_shapes (point, shapes); return 8;
}
return 0;
}
/* finite element shape derivatives; returns the number of nodes */
inline static int element_derivs (int type, double *point, double *derivs)
{
switch (type)
{
case 4: tet_o1_derivs (point, derivs); return 4;
case 5: pyr_o1_derivs (point, derivs); return 5;
case 6: wed_o1_derivs (point, derivs); return 6;
case 8: hex_o1_derivs (point, derivs); return 8;
}
return 0;
}
/* copy element node coordinates into a local table */
inline static int element_nodes (node_t heap, int type, int *nodes, node_t stack)
{
int i;
for (i = 0; i < type; i ++)
{
COPY (heap [nodes [i]], stack [i]);
}
return i;
}
/* copy element nodal values into a local table */
static int element_nodal_values (double *heap, ELEMENT *ele, double (*q) [3])
{
double *p;
int i;
for (i = 0; i < ele->type; i ++)
{
p = &heap [3 * ele->nodes [i]];
COPY (p, q[i]);
}
return i;
}
/* copy element node numbers into a local table */
static int element_node_numbers (ELEMENT *ele, int *numbers)
{
int i;
for (i = 0; i < ele->type; i ++) numbers [i] = ele->nodes [i];
return i;
}
/* local to global point transformation */
static void local_to_global (node_t heap, ELEMENT *ele, double *local, double *global)
{
double nodes [MAX_NODES][3], shapes [MAX_NODES];
int i, j, n;
n = element_nodes (heap, ele->type, ele->nodes, nodes);
element_shapes (ele->type, local, shapes);
SET (global, 0.0);
for (i = 0; i < 3; i ++)
for (j = 0; j < n; j ++)
global [i] += nodes [j][i] * shapes [j];
}
/* global to local point transformation */
static void global_to_local (node_t heap, ELEMENT *ele, double *global, double *local)
{
double nodes [MAX_NODES][3], derivs [3*MAX_NODES], shapes [MAX_NODES];
double A [9], B [3], error;
int i, j, k, l, n, ipiv [3];
element_nodes (heap, ele->type, ele->nodes, nodes);
SET (local, 0.0);
l = 0;
do
{
n = element_shapes (ele->type, local, shapes);
COPY (global, B);
for (i = 0; i < 3; i ++)
for (j = 0; j < n; j ++)
B [i] -= nodes [j][i] * shapes [j];
element_derivs (ele->type, local, derivs);
SET9 (A, 0.0);
for (i = 0; i < 3; i ++)
for (j = 0; j < 3; j ++)
for (k = 0; k < n; k ++) A [3*j+i] += nodes[k][i] * derivs [3*k+j];
ASSERT (lapack_dgesv (3, 1, A, 3, ipiv, B, 3) == 0, ERR_FEM_COORDS_INVERT);
ADD (local, B, local);
error = LEN (B);
} while (++l < 64 && error > 1E-9);
ASSERT (l < 64, ERR_FEM_COORDS_INVERT);
for (i = 0; i < 3; i ++) /* project back onto the element if the point is off due to roundoff */
{
if (local [i] < mincoord [ele->type][i]) local [i] = mincoord [ele->type][i];
else if (local [i] > maxcoord [ele->type][i]) local [i] = maxcoord [ele->type][i];
}
}
/* compute local coordinates of a spatial point */
#define spatial_to_local(msh, ele, x, point) global_to_local ((msh)->cur_nodes, ele, x, point)
/* compute local coordinates of a referential point */
#define referential_to_local(msh, ele, X, point) global_to_local ((msh)->ref_nodes, ele, X, point)
/* compute spatial coordinates of a local point */
#define local_to_spatial(msh, ele, x, point) local_to_global ((msh)->cur_nodes, ele, x, point)
/* compute referential coordinates of a local point */
#define local_to_referential(msh, ele, X, point) local_to_global ((msh)->ref_nodes, ele, X, point)
/* coordinates transformation determinant at local point */
inline static double element_det (int type, node_t nodes, double *point, double *F)
{
double derivs [3*MAX_NODES], G [9];
int i, j, k, n;
n = element_derivs (type, point, derivs);
if (!F) F = G;
SET9 (F, 0.0);
for (i = 0; i < 3; i ++)
for (j = 0; j < 3; j ++)
for (k = 0; k < n; k ++) F [3*j+i] += nodes[k][i] * derivs [3*k+j];
return DET (F);
}
/* element deformation determinant at local point */
inline static void element_gradient (int type, node_t q, double *point, double *F0, double *derivs, double *F)
{
double local_derivs [3*MAX_NODES], IF0 [9], det, *l, *d;
int i, j, k, n;
n = element_derivs (type, point, local_derivs);
INVERT (F0, IF0, det);
ASSERT (det > 0.0, ERR_FEM_COORDS_INVERT);
for (k = 0, l = local_derivs, d = derivs; k < n; k ++, l += 3, d += 3) { TVMUL (IF0, l, d); }
IDENTITY (F);
for (i = 0; i < 3; i ++)
for (j = 0; j < 3; j ++)
for (k = 0; k < n; k ++) F [3*j+i] += q[k][i] * derivs [3*k+j];
}
/* compute element shape functions at a local point and return global matrix */
static MX* element_shapes_matrix (BODY *bod, MESH *msh, ELEMENT *ele, double *point)
{
static int *p, *i, *q, *u, k, n, m, o;
double shapes [MAX_NODES], *x, *y;
int dofs = MESH_DOFS (msh);
MX *N;
o = element_shapes (ele->type, point, shapes);
n = dofs;
ERRMEM (p = MEM_CALLOC ((2*n+1) * sizeof (int)));
i = p + n + 1;
for (k = 0, u = i; k < o; k ++, u += 3)
{
m = ele->nodes [k] * 3;
q = &p [m + 1];
q [0] = 1; q [1] = 1; q [2] = 1;
u [0] = 0; u [1] = 1; u [2] = 2;
}
for (q = p + 1; q < i; q ++) *q += *(q-1);
N = MX_Create (MXCSC, 3, n, p, i);
x = N->x;
free (p);
p = N->p;
for (k = 0; k < o; k ++)
{
m = ele->nodes [k] * 3;
y = &x [p [m]];
SET (y, shapes [k]);
}
return N;
}
/* return number of integration points for internal force computation */
static int number_of_integration_points (ELEMENT *ele)
{
int i = 0;
INTEGRATE3D (ele->type, INTF, ele->dom, ele->domnum, i ++;)
return i;
}
/* allocate bulk material states at integration points */
static void allocate_element_states (MESH *msh, BULK_MATERIAL *mat)
{
ELEMENT *ele;
int nip;
for (ele = msh->surfeles; ele; ele = ele->next)
{
if (mat->nstate && ele->state == NULL)
{
nip = number_of_integration_points (ele);
ERRMEM (ele->state = MEM_CALLOC (nip * sizeof (double [mat->nstate])));
}
}
}
/* simplex integrated volume ==
* shape functions integrated volume test */
static void test_volume_integral (MESH *msh, double ref_volume, int body_id)
{
double J, volume, nodes [MAX_NODES][3];
ELEMENT *ele;
int bulk;
for (ele = msh->surfeles, bulk = 0, volume = 0.0; ele; )
{
element_nodes (msh->ref_nodes, ele->type, ele->nodes, nodes);
INTEGRATE3D (ele->type, MASS, ele->dom, ele->domnum,
J = element_det (ele->type, nodes, point, NULL);
volume += J * weight;
)
if (bulk) ele = ele->next;
else if (ele->next) ele = ele->next;
else ele = msh->bulkeles, bulk = 1;
}
WARNING (fabs (volume - ref_volume) < CUT_TOL * ref_volume,
"FEM BODY %d:\nShape volume is %g.\nIntegrated volume is %g.\n"
"Error %g is beyond the tolerance of %g.\n"
"This issue occurs when your background hexahedrons are not rectilinear.\n"
"Refine the background mesh or make it rectilinear. Alternately, use a tetrahedral background mesh.\n",
body_id, ref_volume, volume, fabs (ref_volume - volume) / ref_volume, CUT_TOL);
}
/* lump contribution of the element mass into the global diagonal matrix */
static void element_lump_mass (BODY *bod, MESH *msh, ELEMENT *ele, double *x)
{
double J, coef, integral, density,
nodes [MAX_NODES][3],
shapes [MAX_NODES],
*out [MAX_NODES];
int i, j, n;
density = ele->mat ? ele->mat->density : bod->mat->density;
n = element_nodes (msh->ref_nodes, ele->type, ele->nodes, nodes);
for (i = 0; i < n; i ++) out [i] = &x [3 * ele->nodes [i]];
INTEGRATE3D (ele->type, MASS, ele->dom, ele->domnum,
element_shapes (ele->type, point, shapes);
J = element_det (ele->type, nodes, point, NULL);