implicit_blender.cc

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/* SPDX-FileCopyrightText: Blender Authors
 *
 * SPDX-License-Identifier: GPL-2.0-or-later */

 
#include "implicit.h"
 
#ifdef IMPLICIT_SOLVER_BLENDER
 
#  include "MEM_guardedalloc.h"
 
#  include "BLI_math_geom.h"
#  include "BLI_math_matrix.h"
#  include "BLI_math_vector.h"
#  include "BLI_task.hh"
 
#  include "BKE_cloth.hh"
 
#  include "SIM_mass_spring.h"
 
#  ifdef __GNUC__
#    pragma GCC diagnostic ignored "-Wtype-limits"
#  endif
 
#  define CLOTH_PARALLEL_LIMIT 1024
 
// #define DEBUG_TIME
 
#  ifdef DEBUG_TIME
#    include "BLI_time.h"
#  endif
 
static float I[3][3] = {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}};
static float ZERO[3][3] = {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}};
 
#  if 0
#    define C99
#    ifdef C99
#      defineDO_INLINE inline
#    else
#      defineDO_INLINE static
#    endif
#  endif /* if 0 */

/* fast vector / matrix library, enhancements are welcome :) -dg */
 
/* DEFINITIONS */
using lfVector = float[3];
struct fmatrix3x3 {
  float m[3][3]; /* 3x3 matrix */
  uint c, r;     /* column and row number */
  // int pinned; /* is this vertex allowed to move? */
  float n1, n2, n3; /* three normal vectors for collision constrains */
  uint vcount;      /* vertex count */
  uint scount;      /* spring count */
};

/* float[3] vector */
/* simple vector code */
/* STATUS: verified */
DO_INLINE void mul_fvector_S(float to[3], const float from[3], float scalar)
{
  to[0] = from[0] * scalar;
  to[1] = from[1] * scalar;
  to[2] = from[2] * scalar;
}
/* simple v^T * v product ("outer product") */
/* STATUS: HAS TO BE verified (*should* work) */
DO_INLINE void mul_fvectorT_fvector(float to[3][3], const float vectorA[3], const float vectorB[3])
{
  mul_fvector_S(to[0], vectorB, vectorA[0]);
  mul_fvector_S(to[1], vectorB, vectorA[1]);
  mul_fvector_S(to[2], vectorB, vectorA[2]);
}
/* simple v^T * v product with scalar ("outer product") */
/* STATUS: HAS TO BE verified (*should* work) */
DO_INLINE void mul_fvectorT_fvectorS(float to[3][3], float vectorA[3], float vectorB[3], float aS)
{
  mul_fvectorT_fvector(to, vectorA, vectorB);
 
  mul_fvector_S(to[0], to[0], aS);
  mul_fvector_S(to[1], to[1], aS);
  mul_fvector_S(to[2], to[2], aS);
}
 
#  if 0
/* printf vector[3] on console: for debug output */
static void print_fvector(float m3[3])
{
  printf("%f\n%f\n%f\n\n", m3[0], m3[1], m3[2]);
}

/* long float vector float (*)[3] */
/* print long vector on console: for debug output */
DO_INLINE void print_lfvector(float (*fLongVector)[3], uint verts)
{
  uint i = 0;
  for (i = 0; i < verts; i++) {
    print_fvector(fLongVector[i]);
  }
}
#  endif
 
/* create long vector */
DO_INLINE lfVector *create_lfvector(uint verts)
{
  /* TODO: check if memory allocation was successful */
  return MEM_calloc_arrayN<lfVector>(verts, "cloth_implicit_alloc_vector");
  // return (lfVector *)cloth_aligned_malloc(&MEMORY_BASE, verts * sizeof(lfVector));
}
/* delete long vector */
DO_INLINE void del_lfvector(float (*fLongVector)[3])
{
  if (fLongVector != nullptr) {
    MEM_freeN(fLongVector);
    // cloth_aligned_free(&MEMORY_BASE, fLongVector);
  }
}
/* copy long vector */
DO_INLINE void cp_lfvector(float (*to)[3], float (*from)[3], uint verts)
{
  memcpy(to, from, verts * sizeof(lfVector));
}
/* init long vector with float[3] */
DO_INLINE void init_lfvector(float (*fLongVector)[3], const float vector[3], uint verts)
{
  uint i = 0;
  for (i = 0; i < verts; i++) {
    copy_v3_v3(fLongVector[i], vector);
  }
}
/* zero long vector with float[3] */
DO_INLINE void zero_lfvector(float (*to)[3], uint verts)
{
  memset(to, 0.0f, verts * sizeof(lfVector));
}
/* Multiply long vector with scalar. */
DO_INLINE void mul_lfvectorS(float (*to)[3], float (*fLongVector)[3], float scalar, uint verts)
{
  uint i = 0;
 
  for (i = 0; i < verts; i++) {
    mul_fvector_S(to[i], fLongVector[i], scalar);
  }
}
/* Multiply long vector with scalar.
 * `A -= B * float` */
DO_INLINE void submul_lfvectorS(float (*to)[3], float (*fLongVector)[3], float scalar, uint verts)
{
  uint i = 0;
  for (i = 0; i < verts; i++) {
    VECSUBMUL(to[i], fLongVector[i], scalar);
  }
}
/* dot product for big vector */
DO_INLINE float dot_lfvector(float (*fLongVectorA)[3], float (*fLongVectorB)[3], uint verts)
{
#  if 0
  /* TODO: try enabling this and measuring performance. It was previously disabled
   * due to non-deterministic behavior, but parallel_deterministic_reduce should
   * give consistent results. */
  return blender::threading::parallel_deterministic_reduce(
      blender::IndexRange(0, verts),
      CLOTH_PARALLEL_LIMIT,
      0.0,
      [=](const blender::IndexRange &range, float value) {
        float temp = value;
        for (const int i : range) {
          temp += dot_v3v3(fLongVectorA[i], fLongVectorB[i]);
        }
        return temp;
      },
      std::plus<>());
#  else
  float temp = 0.0;
  for (uint i = 0; i < verts; i++) {
    temp += dot_v3v3(fLongVectorA[i], fLongVectorB[i]);
  }
  return temp;
#  endif
}
/* `A = B + C` -> for big vector. */
DO_INLINE void add_lfvector_lfvector(float (*to)[3],
                                     float (*fLongVectorA)[3],
                                     float (*fLongVectorB)[3],
                                     uint verts)
{
  uint i = 0;
 
  for (i = 0; i < verts; i++) {
    add_v3_v3v3(to[i], fLongVectorA[i], fLongVectorB[i]);
  }
}
/* `A = B + C * float` -> for big vector. */
DO_INLINE void add_lfvector_lfvectorS(
    float (*to)[3], float (*fLongVectorA)[3], float (*fLongVectorB)[3], float bS, uint verts)
{
  uint i = 0;
 
  for (i = 0; i < verts; i++) {
    VECADDS(to[i], fLongVectorA[i], fLongVectorB[i], bS);
  }
}
/* `A = B * float + C * float` -> for big vector */
DO_INLINE void add_lfvectorS_lfvectorS(float (*to)[3],
                                       float (*fLongVectorA)[3],
                                       float aS,
                                       float (*fLongVectorB)[3],
                                       float bS,
                                       uint verts)
{
  uint i = 0;
 
  for (i = 0; i < verts; i++) {
    VECADDSS(to[i], fLongVectorA[i], aS, fLongVectorB[i], bS);
  }
}
/* `A = B - C * float` -> for big vector. */
DO_INLINE void sub_lfvector_lfvectorS(
    float (*to)[3], float (*fLongVectorA)[3], float (*fLongVectorB)[3], float bS, uint verts)
{
  uint i = 0;
  for (i = 0; i < verts; i++) {
    VECSUBS(to[i], fLongVectorA[i], fLongVectorB[i], bS);
  }
}
/* `A = B - C` -> for big vector. */
DO_INLINE void sub_lfvector_lfvector(float (*to)[3],
                                     float (*fLongVectorA)[3],
                                     float (*fLongVectorB)[3],
                                     uint verts)
{
  uint i = 0;
 
  for (i = 0; i < verts; i++) {
    sub_v3_v3v3(to[i], fLongVectorA[i], fLongVectorB[i]);
  }
}

// 3x3 matrix
#  if 0
/* printf 3x3 matrix on console: for debug output */
static void print_fmatrix(float m3[3][3])
{
  printf("%f\t%f\t%f\n", m3[0][0], m3[0][1], m3[0][2]);
  printf("%f\t%f\t%f\n", m3[1][0], m3[1][1], m3[1][2]);
  printf("%f\t%f\t%f\n\n", m3[2][0], m3[2][1], m3[2][2]);
}
 
static void print_sparse_matrix(fmatrix3x3 *m)
{
  if (m) {
    uint i;
    for (i = 0; i < m[0].vcount + m[0].scount; i++) {
      printf("%d:\n", i);
      print_fmatrix(m[i].m);
    }
  }
}
#  endif
 
#  if 0
static void print_lvector(lfVector *v, int numverts)
{
  int i;
  for (i = 0; i < numverts; i++) {
    if (i > 0) {
      printf("\n");
    }
 
    printf("%f,\n", v[i][0]);
    printf("%f,\n", v[i][1]);
    printf("%f,\n", v[i][2]);
  }
}
#  endif
 
#  if 0
static void print_bfmatrix(fmatrix3x3 *m)
{
  int tot = m[0].vcount + m[0].scount;
  int size = m[0].vcount * 3;
  float *t = MEM_calloc_array<float>N(size * size, "bfmatrix");
  int q, i, j;
 
  for (q = 0; q < tot; q++) {
    int k = 3 * m[q].r;
    int l = 3 * m[q].c;
 
    for (j = 0; j < 3; j++) {
      for (i = 0; i < 3; i++) {
#    if 0
        if (t[k + i + (l + j) * size] != 0.0f) {
          printf("warning: overwriting value at %d, %d\n", m[q].r, m[q].c);
        }
#    endif
        if (k == l) {
          t[k + i + (k + j) * size] += m[q].m[i][j];
        }
        else {
          t[k + i + (l + j) * size] += m[q].m[i][j];
          t[l + j + (k + i) * size] += m[q].m[j][i];
        }
      }
    }
  }
 
  for (j = 0; j < size; j++) {
    if (j > 0 && j % 3 == 0) {
      printf("\n");
    }
 
    for (i = 0; i < size; i++) {
      if (i > 0 && i % 3 == 0) {
        printf("  ");
      }
 
      implicit_print_matrix_elem(t[i + j * size]);
    }
    printf("\n");
  }
 
  MEM_freeN(t);
}
#  endif
 
/* copy 3x3 matrix */
DO_INLINE void cp_fmatrix(float to[3][3], const float from[3][3])
{
  // memcpy(to, from, sizeof(float[3][3]));
  copy_v3_v3(to[0], from[0]);
  copy_v3_v3(to[1], from[1]);
  copy_v3_v3(to[2], from[2]);
}
 
/* copy 3x3 matrix */
DO_INLINE void initdiag_fmatrixS(float to[3][3], float aS)
{
  cp_fmatrix(to, ZERO);
 
  to[0][0] = aS;
  to[1][1] = aS;
  to[2][2] = aS;
}
 
#  if 0
/* calculate determinant of 3x3 matrix */
DO_INLINE float det_fmatrix(float m[3][3])
{
  return m[0][0] * m[1][1] * m[2][2] + m[1][0] * m[2][1] * m[0][2] + m[0][1] * m[1][2] * m[2][0] -
         m[0][0] * m[1][2] * m[2][1] - m[0][1] * m[1][0] * m[2][2] - m[2][0] * m[1][1] * m[0][2];
}
 
DO_INLINE void inverse_fmatrix(float to[3][3], float from[3][3])
{
  uint i, j;
  float d;
 
  if ((d = det_fmatrix(from)) == 0) {
    // printf("can't build inverse");
  }
  for (i = 0; i < 3; i++) {
    for (j = 0; j < 3; j++) {
      int i1 = (i + 1) % 3;
      int i2 = (i + 2) % 3;
      int j1 = (j + 1) % 3;
      int j2 = (j + 2) % 3;
      to[j][i] = (from[i1][j1] * from[i2][j2] - from[i1][j2] * from[i2][j1]) / d;
    }
  }
}
#  endif
 
/* 3x3 matrix multiplied by a scalar */
/* STATUS: verified */
DO_INLINE void mul_fmatrix_S(float matrix[3][3], float scalar)
{
  mul_fvector_S(matrix[0], matrix[0], scalar);
  mul_fvector_S(matrix[1], matrix[1], scalar);
  mul_fvector_S(matrix[2], matrix[2], scalar);
}
 
/* a vector multiplied by a 3x3 matrix */
/* STATUS: verified */
DO_INLINE void mul_fvector_fmatrix(float *to, const float *from, const float matrix[3][3])
{
  to[0] = matrix[0][0] * from[0] + matrix[1][0] * from[1] + matrix[2][0] * from[2];
  to[1] = matrix[0][1] * from[0] + matrix[1][1] * from[1] + matrix[2][1] * from[2];
  to[2] = matrix[0][2] * from[0] + matrix[1][2] * from[1] + matrix[2][2] * from[2];
}
 
/* 3x3 matrix multiplied by a vector */
/* STATUS: verified */
DO_INLINE void mul_fmatrix_fvector(float *to, const float matrix[3][3], const float from[3])
{
  to[0] = dot_v3v3(matrix[0], from);
  to[1] = dot_v3v3(matrix[1], from);
  to[2] = dot_v3v3(matrix[2], from);
}
/* 3x3 matrix addition with 3x3 matrix */
DO_INLINE void add_fmatrix_fmatrix(float to[3][3],
                                   const float matrixA[3][3],
                                   const float matrixB[3][3])
{
  add_v3_v3v3(to[0], matrixA[0], matrixB[0]);
  add_v3_v3v3(to[1], matrixA[1], matrixB[1]);
  add_v3_v3v3(to[2], matrixA[2], matrixB[2]);
}
/* `A -= B*x + (C * y)` (3x3 matrix sub-addition with 3x3 matrix). */
DO_INLINE void subadd_fmatrixS_fmatrixS(
    float to[3][3], const float matrixA[3][3], float aS, const float matrixB[3][3], float bS)
{
  VECSUBADDSS(to[0], matrixA[0], aS, matrixB[0], bS);
  VECSUBADDSS(to[1], matrixA[1], aS, matrixB[1], bS);
  VECSUBADDSS(to[2], matrixA[2], aS, matrixB[2], bS);
}
/* `A = B - C` (3x3 matrix subtraction with 3x3 matrix). */
DO_INLINE void sub_fmatrix_fmatrix(float to[3][3],
                                   const float matrixA[3][3],
                                   const float matrixB[3][3])
{
  sub_v3_v3v3(to[0], matrixA[0], matrixB[0]);
  sub_v3_v3v3(to[1], matrixA[1], matrixB[1]);
  sub_v3_v3v3(to[2], matrixA[2], matrixB[2]);
}

/* special functions */
/* 3x3 matrix multiplied+added by a vector */
/* STATUS: verified */
DO_INLINE void muladd_fmatrix_fvector(float to[3], const float matrix[3][3], const float from[3])
{
  to[0] += dot_v3v3(matrix[0], from);
  to[1] += dot_v3v3(matrix[1], from);
  to[2] += dot_v3v3(matrix[2], from);
}
 
DO_INLINE void muladd_fmatrixT_fvector(float to[3], const float matrix[3][3], const float from[3])
{
  to[0] += matrix[0][0] * from[0] + matrix[1][0] * from[1] + matrix[2][0] * from[2];
  to[1] += matrix[0][1] * from[0] + matrix[1][1] * from[1] + matrix[2][1] * from[2];
  to[2] += matrix[0][2] * from[0] + matrix[1][2] * from[1] + matrix[2][2] * from[2];
}
 
BLI_INLINE void outerproduct(float r[3][3], const float a[3], const float b[3])
{
  mul_v3_v3fl(r[0], a, b[0]);
  mul_v3_v3fl(r[1], a, b[1]);
  mul_v3_v3fl(r[2], a, b[2]);
}
 
BLI_INLINE void cross_m3_v3m3(float r[3][3], const float v[3], const float m[3][3])
{
  cross_v3_v3v3(r[0], v, m[0]);
  cross_v3_v3v3(r[1], v, m[1]);
  cross_v3_v3v3(r[2], v, m[2]);
}
 
BLI_INLINE void cross_v3_identity(float r[3][3], const float v[3])
{
  r[0][0] = 0.0f;
  r[1][0] = v[2];
  r[2][0] = -v[1];
  r[0][1] = -v[2];
  r[1][1] = 0.0f;
  r[2][1] = v[0];
  r[0][2] = v[1];
  r[1][2] = -v[0];
  r[2][2] = 0.0f;
}
 
BLI_INLINE void madd_m3_m3fl(float r[3][3], const float m[3][3], float f)
{
  r[0][0] += m[0][0] * f;
  r[0][1] += m[0][1] * f;
  r[0][2] += m[0][2] * f;
  r[1][0] += m[1][0] * f;
  r[1][1] += m[1][1] * f;
  r[1][2] += m[1][2] * f;
  r[2][0] += m[2][0] * f;
  r[2][1] += m[2][1] * f;
  r[2][2] += m[2][2] * f;
}


/* SPARSE SYMMETRIC big matrix with 3x3 matrix entries */
/* printf a big matrix on console: for debug output */
#  if 0
static void print_bfmatrix(fmatrix3x3 *m3)
{
  uint i = 0;
 
  for (i = 0; i < m3[0].vcount + m3[0].scount; i++) {
    print_fmatrix(m3[i].m);
  }
}
#  endif
 
BLI_INLINE void init_fmatrix(fmatrix3x3 *matrix, int r, int c)
{
  matrix->r = r;
  matrix->c = c;
}
 
/* create big matrix */
DO_INLINE fmatrix3x3 *create_bfmatrix(uint verts, uint springs)
{
  /* TODO: check if memory allocation was successful */
  fmatrix3x3 *temp = MEM_calloc_arrayN<fmatrix3x3>(verts + springs, "cloth_implicit_alloc_matrix");
  int i;
 
  temp[0].vcount = verts;
  temp[0].scount = springs;
 
  /* vertex part of the matrix is diagonal blocks */
  for (i = 0; i < verts; i++) {
    init_fmatrix(temp + i, i, i);
  }
 
  return temp;
}
/* delete big matrix */
DO_INLINE void del_bfmatrix(fmatrix3x3 *matrix)
{
  if (matrix != nullptr) {
    MEM_freeN(matrix);
  }
}
 
/* copy big matrix */
DO_INLINE void cp_bfmatrix(fmatrix3x3 *to, fmatrix3x3 *from)
{
  /* TODO: bounds checking. */
  memcpy(to, from, sizeof(fmatrix3x3) * (from[0].vcount + from[0].scount));
}
 
/* init big matrix */
/* slow in parallel */
DO_INLINE void init_bfmatrix(fmatrix3x3 *matrix, float m3[3][3])
{
  uint i;
 
  for (i = 0; i < matrix[0].vcount + matrix[0].scount; i++) {
    cp_fmatrix(matrix[i].m, m3);
  }
}
 
/* init the diagonal of big matrix */
/* slow in parallel */
DO_INLINE void initdiag_bfmatrix(fmatrix3x3 *matrix, float m3[3][3])
{
  uint i, j;
  float tmatrix[3][3] = {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}};
 
  for (i = 0; i < matrix[0].vcount; i++) {
    cp_fmatrix(matrix[i].m, m3);
  }
  for (j = matrix[0].vcount; j < matrix[0].vcount + matrix[0].scount; j++) {
    cp_fmatrix(matrix[j].m, tmatrix);
  }
}
 
/* SPARSE SYMMETRIC multiply big matrix with long vector. */
/* STATUS: verified */
DO_INLINE void mul_bfmatrix_lfvector(float (*to)[3], fmatrix3x3 *from, lfVector *fLongVector)
{
  uint vcount = from[0].vcount;
  lfVector *temp = create_lfvector(vcount);
 
  zero_lfvector(to, vcount);
 
  blender::threading::parallel_invoke(
      vcount > CLOTH_PARALLEL_LIMIT,
      [&]() {
        for (uint i = from[0].vcount; i < from[0].vcount + from[0].scount; i++) {
          /* This is the lower triangle of the sparse matrix,
           * therefore multiplication occurs with transposed sub-matrices. */
          muladd_fmatrixT_fvector(to[from[i].c], from[i].m, fLongVector[from[i].r]);
        }
      },
      [&]() {
        for (uint i = 0; i < from[0].vcount + from[0].scount; i++) {
          muladd_fmatrix_fvector(temp[from[i].r], from[i].m, fLongVector[from[i].c]);
        }
      });
 
  add_lfvector_lfvector(to, to, temp, from[0].vcount);
 
  del_lfvector(temp);
}
 
/* SPARSE SYMMETRIC sub big matrix with big matrix. */
/* A -= B * float + C * float --> for big matrix */
/* VERIFIED */
DO_INLINE void subadd_bfmatrixS_bfmatrixS(
    fmatrix3x3 *to, fmatrix3x3 *from, float aS, fmatrix3x3 *matrix, float bS)
{
  uint i = 0;
 
  /* process diagonal elements */
  for (i = 0; i < matrix[0].vcount + matrix[0].scount; i++) {
    subadd_fmatrixS_fmatrixS(to[i].m, from[i].m, aS, matrix[i].m, bS);
  }
}

/* simulator start */
 
struct Implicit_Data {
  /* inputs */
  fmatrix3x3 *bigI;        /* identity (constant) */
  fmatrix3x3 *tfm;         /* local coordinate transform */
  fmatrix3x3 *M;           /* masses */
  lfVector *F;             /* forces */
  fmatrix3x3 *dFdV, *dFdX; /* force jacobians */
  int num_blocks;          /* number of off-diagonal blocks (springs) */
 
  /* motion state data */
  lfVector *X, *Xnew; /* positions */
  lfVector *V, *Vnew; /* velocities */
 
  /* internal solver data */
  lfVector *B;   /* B for A*dV = B */
  fmatrix3x3 *A; /* A for A*dV = B */
 
  lfVector *dV;         /* velocity change (solution of A*dV = B) */
  lfVector *z;          /* target velocity in constrained directions */
  fmatrix3x3 *S;        /* filtering matrix for constraints */
  fmatrix3x3 *P, *Pinv; /* pre-conditioning matrix */
};
 
Implicit_Data *SIM_mass_spring_solver_create(int numverts, int numsprings)
{
  Implicit_Data *id = MEM_callocN<Implicit_Data>("implicit vecmat");
 
  /* process diagonal elements */
  id->tfm = create_bfmatrix(numverts, 0);
  id->A = create_bfmatrix(numverts, numsprings);
  id->dFdV = create_bfmatrix(numverts, numsprings);
  id->dFdX = create_bfmatrix(numverts, numsprings);
  id->S = create_bfmatrix(numverts, 0);
  id->Pinv = create_bfmatrix(numverts, numsprings);
  id->P = create_bfmatrix(numverts, numsprings);
  id->bigI = create_bfmatrix(numverts, numsprings); /* TODO: 0 springs. */
  id->M = create_bfmatrix(numverts, numsprings);
  id->X = create_lfvector(numverts);
  id->Xnew = create_lfvector(numverts);
  id->V = create_lfvector(numverts);
  id->Vnew = create_lfvector(numverts);
  id->F = create_lfvector(numverts);
  id->B = create_lfvector(numverts);
  id->dV = create_lfvector(numverts);
  id->z = create_lfvector(numverts);
 
  initdiag_bfmatrix(id->bigI, I);
 
  return id;
}
 
void SIM_mass_spring_solver_free(Implicit_Data *id)
{
  del_bfmatrix(id->tfm);
  del_bfmatrix(id->A);
  del_bfmatrix(id->dFdV);
  del_bfmatrix(id->dFdX);
  del_bfmatrix(id->S);
  del_bfmatrix(id->P);
  del_bfmatrix(id->Pinv);
  del_bfmatrix(id->bigI);
  del_bfmatrix(id->M);
 
  del_lfvector(id->X);
  del_lfvector(id->Xnew);
  del_lfvector(id->V);
  del_lfvector(id->Vnew);
  del_lfvector(id->F);
  del_lfvector(id->B);
  del_lfvector(id->dV);
  del_lfvector(id->z);
 
  MEM_freeN(id);
}
 
/* ==== Transformation from/to root reference frames ==== */
 
BLI_INLINE void world_to_root_v3(Implicit_Data *data, int index, float r[3], const float v[3])
{
  copy_v3_v3(r, v);
  mul_transposed_m3_v3(data->tfm[index].m, r);
}
 
BLI_INLINE void root_to_world_v3(Implicit_Data *data, int index, float r[3], const float v[3])
{
  mul_v3_m3v3(r, data->tfm[index].m, v);
}
 
BLI_INLINE void world_to_root_m3(Implicit_Data *data,
                                 int index,
                                 float r[3][3],
                                 const float m[3][3])
{
  float trot[3][3];
  copy_m3_m3(trot, data->tfm[index].m);
  transpose_m3(trot);
  mul_m3_m3m3(r, trot, m);
}
 
BLI_INLINE void root_to_world_m3(Implicit_Data *data,
                                 int index,
                                 float r[3][3],
                                 const float m[3][3])
{
  mul_m3_m3m3(r, data->tfm[index].m, m);
}
 
/* ================================ */
 
DO_INLINE void filter(lfVector *V, fmatrix3x3 *S)
{
  uint i = 0;
 
  for (i = 0; i < S[0].vcount; i++) {
    mul_m3_v3(S[i].m, V[S[i].r]);
  }
}
 
/* this version of the CG algorithm does not work very well with partial constraints
 * (where S has non-zero elements). */
#  if 0
static int cg_filtered(lfVector *ldV, fmatrix3x3 *lA, lfVector *lB, lfVector *z, fmatrix3x3 *S)
{
  /* Solves for unknown X in equation AX=B */
  uint conjgrad_loopcount = 0, conjgrad_looplimit = 100;
  float conjgrad_epsilon = 0.0001f /* , conjgrad_lasterror=0 */ /* UNUSED */;
  lfVector *q, *d, *tmp, *r;
  float s, starget, a, s_prev;
  uint numverts = lA[0].vcount;
  q = create_lfvector(numverts);
  d = create_lfvector(numverts);
  tmp = create_lfvector(numverts);
  r = create_lfvector(numverts);
 
  // zero_lfvector(ldV, CLOTHPARTICLES);
  filter(ldV, S);
 
  add_lfvector_lfvector(ldV, ldV, z, numverts);
 
  // r = B - Mul(tmp, A, X);    /* just use B if X known to be zero. */
  cp_lfvector(r, lB, numverts);
  mul_bfmatrix_lfvector(tmp, lA, ldV);
  sub_lfvector_lfvector(r, r, tmp, numverts);
 
  filter(r, S);
 
  cp_lfvector(d, r, numverts);
 
  s = dot_lfvector(r, r, numverts);
  starget = s * sqrtf(conjgrad_epsilon);
 
  while (s > starget && conjgrad_loopcount < conjgrad_looplimit) {
    // Mul(q, A, d); /* q = A*d; */
    mul_bfmatrix_lfvector(q, lA, d);
 
    filter(q, S);
 
    a = s / dot_lfvector(d, q, numverts);
 
    /* `X = X + d*a;` */
    add_lfvector_lfvectorS(ldV, ldV, d, a, numverts);
 
    /* `r = r - q*a;` */
    sub_lfvector_lfvectorS(r, r, q, a, numverts);
 
    s_prev = s;
    s = dot_lfvector(r, r, numverts);
 
    /* `d = r+d*(s/s_prev);` */
    add_lfvector_lfvectorS(d, r, d, (s / s_prev), numverts);
 
    filter(d, S);
 
    conjgrad_loopcount++;
  }
  // conjgrad_lasterror = s; /* UNUSED */
 
  del_lfvector(q);
  del_lfvector(d);
  del_lfvector(tmp);
  del_lfvector(r);
  // printf("W/O conjgrad_loopcount: %d\n", conjgrad_loopcount);
 
  /* True means we reached desired accuracy in given time - ie stable. */
  return conjgrad_loopcount < conjgrad_looplimit;
}
#  endif
 
static int cg_filtered(lfVector *ldV,
                       fmatrix3x3 *lA,
                       lfVector *lB,
                       lfVector *z,
                       fmatrix3x3 *S,
                       ImplicitSolverResult *result)
{
  /* Solves for unknown X in equation AX=B */
  uint conjgrad_loopcount = 0, conjgrad_looplimit = 100;
  float conjgrad_epsilon = 0.01f;
 
  uint numverts = lA[0].vcount;
  lfVector *fB = create_lfvector(numverts);
  lfVector *AdV = create_lfvector(numverts);
  lfVector *r = create_lfvector(numverts);
  lfVector *c = create_lfvector(numverts);
  lfVector *q = create_lfvector(numverts);
  lfVector *s = create_lfvector(numverts);
  float bnorm2, delta_new, delta_old, delta_target, alpha;
 
  cp_lfvector(ldV, z, numverts);
 
  /* d0 = filter(B)^T * P * filter(B) */
  cp_lfvector(fB, lB, numverts);
  filter(fB, S);
  bnorm2 = dot_lfvector(fB, fB, numverts);
  delta_target = conjgrad_epsilon * conjgrad_epsilon * bnorm2;
 
  /* r = filter(B - A * dV) */
  mul_bfmatrix_lfvector(AdV, lA, ldV);
  sub_lfvector_lfvector(r, lB, AdV, numverts);
  filter(r, S);
 
  /* c = filter(P^-1 * r) */
  cp_lfvector(c, r, numverts);
  filter(c, S);
 
  /* delta = r^T * c */
  delta_new = dot_lfvector(r, c, numverts);
 
#  ifdef IMPLICIT_PRINT_SOLVER_INPUT_OUTPUT
  printf("==== A ====\n");
  print_bfmatrix(lA);
  printf("==== z ====\n");
  print_lvector(z, numverts);
  printf("==== B ====\n");
  print_lvector(lB, numverts);
  printf("==== S ====\n");
  print_bfmatrix(S);
#  endif
 
  while (delta_new > delta_target && conjgrad_loopcount < conjgrad_looplimit) {
    mul_bfmatrix_lfvector(q, lA, c);
    filter(q, S);
 
    alpha = delta_new / dot_lfvector(c, q, numverts);
 
    add_lfvector_lfvectorS(ldV, ldV, c, alpha, numverts);
 
    add_lfvector_lfvectorS(r, r, q, -alpha, numverts);
 
    /* s = P^-1 * r */
    cp_lfvector(s, r, numverts);
    delta_old = delta_new;
    delta_new = dot_lfvector(r, s, numverts);
 
    add_lfvector_lfvectorS(c, s, c, delta_new / delta_old, numverts);
    filter(c, S);
 
    conjgrad_loopcount++;
  }
 
#  ifdef IMPLICIT_PRINT_SOLVER_INPUT_OUTPUT
  printf("==== dV ====\n");
  print_lvector(ldV, numverts);
  printf("========\n");
#  endif
 
  del_lfvector(fB);
  del_lfvector(AdV);
  del_lfvector(r);
  del_lfvector(c);
  del_lfvector(q);
  del_lfvector(s);
  // printf("W/O conjgrad_loopcount: %d\n", conjgrad_loopcount);
 
  result->status = conjgrad_loopcount < conjgrad_looplimit ? SIM_SOLVER_SUCCESS :
                                                             SIM_SOLVER_NO_CONVERGENCE;
  result->iterations = conjgrad_loopcount;
  result->error = bnorm2 > 0.0f ? sqrtf(delta_new / bnorm2) : 0.0f;
 
  /* True means we reached desired accuracy in given time - ie stable. */
  return conjgrad_loopcount < conjgrad_looplimit;
}
 
#  if 0
/* block diagonalizer */
DO_INLINE void BuildPPinv(fmatrix3x3 *lA, fmatrix3x3 *P, fmatrix3x3 *Pinv)
{
  /* Take only the diagonal blocks of A */
  blender::threading::parallel_for(blender::IndexRange(0, lA[0].vcount),
                                   CLOTH_PARALLEL_LIMIT,
                                   [&](const blender::IndexRange &range) {
                                     for (int64_t i : range) {
                                       /* block diagonalizer */
                                       cp_fmatrix(P[i].m, lA[i].m);
                                       inverse_fmatrix(Pinv[i].m, P[i].m);
                                     }
                                   });
}
 
#    if 0
/* version 1.3 */
static int cg_filtered_pre(lfVector *dv,
                           fmatrix3x3 *lA,
                           lfVector *lB,
                           lfVector *z,
                           fmatrix3x3 *S,
                           fmatrix3x3 *P,
                           fmatrix3x3 *Pinv)
{
  uint numverts = lA[0].vcount, iterations = 0, conjgrad_looplimit = 100;
  float delta0 = 0, deltaNew = 0, deltaOld = 0, alpha = 0;
  float conjgrad_epsilon = 0.0001; /* 0.2 is dt for steps=5 */
  lfVector *r = create_lfvector(numverts);
  lfVector *p = create_lfvector(numverts);
  lfVector *s = create_lfvector(numverts);
  lfVector *h = create_lfvector(numverts);
 
  BuildPPinv(lA, P, Pinv);
 
  filter(dv, S);
  add_lfvector_lfvector(dv, dv, z, numverts);
 
  mul_bfmatrix_lfvector(r, lA, dv);
  sub_lfvector_lfvector(r, lB, r, numverts);
  filter(r, S);
 
  mul_prevfmatrix_lfvector(p, Pinv, r);
  filter(p, S);
 
  deltaNew = dot_lfvector(r, p, numverts);
 
  delta0 = deltaNew * sqrt(conjgrad_epsilon);
 
#      ifdef DEBUG_TIME
  double start = BLI_time_now_seconds();
#      endif
 
  while ((deltaNew > delta0) && (iterations < conjgrad_looplimit)) {
    iterations++;
 
    mul_bfmatrix_lfvector(s, lA, p);
    filter(s, S);
 
    alpha = deltaNew / dot_lfvector(p, s, numverts);
 
    add_lfvector_lfvectorS(dv, dv, p, alpha, numverts);
 
    add_lfvector_lfvectorS(r, r, s, -alpha, numverts);
 
    mul_prevfmatrix_lfvector(h, Pinv, r);
    filter(h, S);
 
    deltaOld = deltaNew;
 
    deltaNew = dot_lfvector(r, h, numverts);
 
    add_lfvector_lfvectorS(p, h, p, deltaNew / deltaOld, numverts);
 
    filter(p, S);
  }
 
#      ifdef DEBUG_TIME
  double end = BLI_time_now_seconds();
  printf("cg_filtered_pre time: %f\n", float(end - start));
#      endif
 
  del_lfvector(h);
  del_lfvector(s);
  del_lfvector(p);
  del_lfvector(r);
 
  printf("iterations: %d\n", iterations);
 
  return iterations < conjgrad_looplimit;
}
#    endif
 
/* version 1.4 */
static int cg_filtered_pre(lfVector *dv,
                           fmatrix3x3 *lA,
                           lfVector *lB,
                           lfVector *z,
                           fmatrix3x3 *S,
                           fmatrix3x3 *P,
                           fmatrix3x3 *Pinv,
                           fmatrix3x3 *bigI)
{
  uint numverts = lA[0].vcount, iterations = 0, conjgrad_looplimit = 100;
  float delta0 = 0, deltaNew = 0, deltaOld = 0, alpha = 0, tol = 0;
  lfVector *r = create_lfvector(numverts);
  lfVector *p = create_lfvector(numverts);
  lfVector *s = create_lfvector(numverts);
  lfVector *h = create_lfvector(numverts);
  lfVector *bhat = create_lfvector(numverts);
  lfVector *btemp = create_lfvector(numverts);
 
  BuildPPinv(lA, P, Pinv);
 
  initdiag_bfmatrix(bigI, I);
  sub_bfmatrix_Smatrix(bigI, bigI, S);
 
  /* x = Sx_0+(I-S)z */
  filter(dv, S);
  add_lfvector_lfvector(dv, dv, z, numverts);
 
  /* b_hat = S(b-A(I-S)z) */
  mul_bfmatrix_lfvector(r, lA, z);
  mul_bfmatrix_lfvector(bhat, bigI, r);
  sub_lfvector_lfvector(bhat, lB, bhat, numverts);
 
  /* r = S(b-Ax) */
  mul_bfmatrix_lfvector(r, lA, dv);
  sub_lfvector_lfvector(r, lB, r, numverts);
  filter(r, S);
 
  /* p = SP^-1r */
  mul_prevfmatrix_lfvector(p, Pinv, r);
  filter(p, S);
 
  /* delta0 = bhat^TP^-1bhat */
  mul_prevfmatrix_lfvector(btemp, Pinv, bhat);
  delta0 = dot_lfvector(bhat, btemp, numverts);
 
  /* deltaNew = r^TP */
  deltaNew = dot_lfvector(r, p, numverts);
 
#    if 0
  filter(dv, S);
  add_lfvector_lfvector(dv, dv, z, numverts);
 
  mul_bfmatrix_lfvector(r, lA, dv);
  sub_lfvector_lfvector(r, lB, r, numverts);
  filter(r, S);
 
  mul_prevfmatrix_lfvector(p, Pinv, r);
  filter(p, S);
 
  deltaNew = dot_lfvector(r, p, numverts);
 
  delta0 = deltaNew * sqrt(conjgrad_epsilon);
#    endif
 
#    ifdef DEBUG_TIME
  double start = BLI_time_now_seconds();
#    endif
 
  tol = (0.01 * 0.2);
 
  while ((deltaNew > delta0 * tol * tol) && (iterations < conjgrad_looplimit)) {
    iterations++;
 
    mul_bfmatrix_lfvector(s, lA, p);
    filter(s, S);
 
    alpha = deltaNew / dot_lfvector(p, s, numverts);
 
    add_lfvector_lfvectorS(dv, dv, p, alpha, numverts);
 
    add_lfvector_lfvectorS(r, r, s, -alpha, numverts);
 
    mul_prevfmatrix_lfvector(h, Pinv, r);
    filter(h, S);
 
    deltaOld = deltaNew;
 
    deltaNew = dot_lfvector(r, h, numverts);
 
    add_lfvector_lfvectorS(p, h, p, deltaNew / deltaOld, numverts);
 
    filter(p, S);
  }
 
#    ifdef DEBUG_TIME
  double end = BLI_time_now_seconds();
  printf("cg_filtered_pre time: %f\n", float(end - start));
#    endif
 
  del_lfvector(btemp);
  del_lfvector(bhat);
  del_lfvector(h);
  del_lfvector(s);
  del_lfvector(p);
  del_lfvector(r);
 
  // printf("iterations: %d\n", iterations);
 
  return iterations < conjgrad_looplimit;
}
#  endif
 
bool SIM_mass_spring_solve_velocities(Implicit_Data *data, float dt, ImplicitSolverResult *result)
{
  uint numverts = data->dFdV[0].vcount;
 
  lfVector *dFdXmV = create_lfvector(numverts);
  zero_lfvector(data->dV, numverts);
 
  cp_bfmatrix(data->A, data->M);
 
  subadd_bfmatrixS_bfmatrixS(data->A, data->dFdV, dt, data->dFdX, (dt * dt));
 
  mul_bfmatrix_lfvector(dFdXmV, data->dFdX, data->V);
 
  add_lfvectorS_lfvectorS(data->B, data->F, dt, dFdXmV, (dt * dt), numverts);
 
#  ifdef DEBUG_TIME
  double start = BLI_time_now_seconds();
#  endif
 
  /* Conjugate gradient algorithm to solve Ax=b. */
  cg_filtered(data->dV, data->A, data->B, data->z, data->S, result);
 
  // cg_filtered_pre(id->dV, id->A, id->B, id->z, id->S, id->P, id->Pinv, id->bigI);
 
#  ifdef DEBUG_TIME
  double end = BLI_time_now_seconds();
  printf("cg_filtered calc time: %f\n", float(end - start));
#  endif
 
  /* advance velocities */
  add_lfvector_lfvector(data->Vnew, data->V, data->dV, numverts);
 
  del_lfvector(dFdXmV);
 
  return result->status == SIM_SOLVER_SUCCESS;
}
 
bool SIM_mass_spring_solve_positions(Implicit_Data *data, float dt)
{
  int numverts = data->M[0].vcount;
 
  /* advance positions */
  add_lfvector_lfvectorS(data->Xnew, data->X, data->Vnew, dt, numverts);
 
  return true;
}
 
void SIM_mass_spring_apply_result(Implicit_Data *data)
{
  int numverts = data->M[0].vcount;
  cp_lfvector(data->X, data->Xnew, numverts);
  cp_lfvector(data->V, data->Vnew, numverts);
}
 
void SIM_mass_spring_set_vertex_mass(Implicit_Data *data, int index, float mass)
{
  unit_m3(data->M[index].m);
  mul_m3_fl(data->M[index].m, mass);
}
 
void SIM_mass_spring_set_rest_transform(Implicit_Data *data, int index, float tfm[3][3])
{
#  ifdef CLOTH_ROOT_FRAME
  copy_m3_m3(data->tfm[index].m, tfm);
#  else
  unit_m3(data->tfm[index].m);
  (void)tfm;
#  endif
}
 
void SIM_mass_spring_set_motion_state(Implicit_Data *data,
                                      int index,
                                      const float x[3],
                                      const float v[3])
{
  world_to_root_v3(data, index, data->X[index], x);
  world_to_root_v3(data, index, data->V[index], v);
}
 
void SIM_mass_spring_set_position(Implicit_Data *data, int index, const float x[3])
{
  world_to_root_v3(data, index, data->X[index], x);
}
 
void SIM_mass_spring_set_velocity(Implicit_Data *data, int index, const float v[3])
{
  world_to_root_v3(data, index, data->V[index], v);
}
 
void SIM_mass_spring_get_motion_state(Implicit_Data *data, int index, float x[3], float v[3])
{
  if (x) {
    root_to_world_v3(data, index, x, data->X[index]);
  }
  if (v) {
    root_to_world_v3(data, index, v, data->V[index]);
  }
}
 
void SIM_mass_spring_get_position(Implicit_Data *data, int index, float x[3])
{
  root_to_world_v3(data, index, x, data->X[index]);
}
 
void SIM_mass_spring_get_velocity(Implicit_Data *data, int index, float v[3])
{
  root_to_world_v3(data, index, v, data->V[index]);
}
 
void SIM_mass_spring_get_new_position(Implicit_Data *data, int index, float x[3])
{
  root_to_world_v3(data, index, x, data->Xnew[index]);
}
 
void SIM_mass_spring_set_new_position(Implicit_Data *data, int index, const float x[3])
{
  world_to_root_v3(data, index, data->Xnew[index], x);
}
 
void SIM_mass_spring_get_new_velocity(Implicit_Data *data, int index, float v[3])
{
  root_to_world_v3(data, index, v, data->Vnew[index]);
}
 
void SIM_mass_spring_set_new_velocity(Implicit_Data *data, int index, const float v[3])
{
  world_to_root_v3(data, index, data->Vnew[index], v);
}
 
/* -------------------------------- */
 
static int SIM_mass_spring_add_block(Implicit_Data *data, int v1, int v2)
{
  int s = data->M[0].vcount + data->num_blocks; /* index from array start */
  BLI_assert(s < data->M[0].vcount + data->M[0].scount);
  ++data->num_blocks;
 
  /* tfm and S don't have spring entries (diagonal blocks only) */
  init_fmatrix(data->bigI + s, v1, v2);
  init_fmatrix(data->M + s, v1, v2);
  init_fmatrix(data->dFdX + s, v1, v2);
  init_fmatrix(data->dFdV + s, v1, v2);
  init_fmatrix(data->A + s, v1, v2);
  init_fmatrix(data->P + s, v1, v2);
  init_fmatrix(data->Pinv + s, v1, v2);
 
  return s;
}
 
void SIM_mass_spring_clear_constraints(Implicit_Data *data)
{
  int i, numverts = data->S[0].vcount;
  for (i = 0; i < numverts; i++) {
    unit_m3(data->S[i].m);
    zero_v3(data->z[i]);
  }
}
 
void SIM_mass_spring_add_constraint_ndof0(Implicit_Data *data, int index, const float dV[3])
{
  zero_m3(data->S[index].m);
 
  world_to_root_v3(data, index, data->z[index], dV);
}
 
void SIM_mass_spring_add_constraint_ndof1(
    Implicit_Data *data, int index, const float c1[3], const float c2[3], const float dV[3])
{
  float m[3][3], p[3], q[3], u[3], cmat[3][3];
 
  world_to_root_v3(data, index, p, c1);
  mul_fvectorT_fvector(cmat, p, p);
  sub_m3_m3m3(m, I, cmat);
 
  world_to_root_v3(data, index, q, c2);
  mul_fvectorT_fvector(cmat, q, q);
  sub_m3_m3m3(m, m, cmat);
 
  /* XXX not sure but multiplication should work here */
  copy_m3_m3(data->S[index].m, m);
  //  mul_m3_m3m3(data->S[index].m, data->S[index].m, m);
 
  world_to_root_v3(data, index, u, dV);
  add_v3_v3(data->z[index], u);
}
 
void SIM_mass_spring_add_constraint_ndof2(Implicit_Data *data,
                                          int index,
                                          const float c1[3],
                                          const float dV[3])
{
  float m[3][3], p[3], u[3], cmat[3][3];
 
  world_to_root_v3(data, index, p, c1);
  mul_fvectorT_fvector(cmat, p, p);
  sub_m3_m3m3(m, I, cmat);
 
  copy_m3_m3(data->S[index].m, m);
  //  mul_m3_m3m3(data->S[index].m, data->S[index].m, m);
 
  world_to_root_v3(data, index, u, dV);
  add_v3_v3(data->z[index], u);
}
 
void SIM_mass_spring_clear_forces(Implicit_Data *data)
{
  int numverts = data->M[0].vcount;
  zero_lfvector(data->F, numverts);
  init_bfmatrix(data->dFdX, ZERO);
  init_bfmatrix(data->dFdV, ZERO);
 
  data->num_blocks = 0;
}
 
void SIM_mass_spring_force_reference_frame(Implicit_Data *data,
                                           int index,
                                           const float acceleration[3],
                                           const float omega[3],
                                           const float domega_dt[3],
                                           float mass)
{
#  ifdef CLOTH_ROOT_FRAME
  float acc[3], w[3], dwdt[3];
  float f[3], dfdx[3][3], dfdv[3][3];
  float euler[3], coriolis[3], centrifugal[3], rotvel[3];
  float deuler[3][3], dcoriolis[3][3], dcentrifugal[3][3], drotvel[3][3];
 
  world_to_root_v3(data, index, acc, acceleration);
  world_to_root_v3(data, index, w, omega);
  world_to_root_v3(data, index, dwdt, domega_dt);
 
  cross_v3_v3v3(euler, dwdt, data->X[index]);
  cross_v3_v3v3(coriolis, w, data->V[index]);
  mul_v3_fl(coriolis, 2.0f);
  cross_v3_v3v3(rotvel, w, data->X[index]);
  cross_v3_v3v3(centrifugal, w, rotvel);
 
  sub_v3_v3v3(f, acc, euler);
  sub_v3_v3(f, coriolis);
  sub_v3_v3(f, centrifugal);
 
  mul_v3_fl(f, mass); /* F = m * a */
 
  cross_v3_identity(deuler, dwdt);
  cross_v3_identity(dcoriolis, w);
  mul_m3_fl(dcoriolis, 2.0f);
  cross_v3_identity(drotvel, w);
  cross_m3_v3m3(dcentrifugal, w, drotvel);
 
  add_m3_m3m3(dfdx, deuler, dcentrifugal);
  negate_m3(dfdx);
  mul_m3_fl(dfdx, mass);
 
  copy_m3_m3(dfdv, dcoriolis);
  negate_m3(dfdv);
  mul_m3_fl(dfdv, mass);
 
  add_v3_v3(data->F[index], f);
  add_m3_m3m3(data->dFdX[index].m, data->dFdX[index].m, dfdx);
  add_m3_m3m3(data->dFdV[index].m, data->dFdV[index].m, dfdv);
#  else
  (void)data;
  (void)index;
  (void)acceleration;
  (void)omega;
  (void)domega_dt;
#  endif
}
 
void SIM_mass_spring_force_gravity(Implicit_Data *data, int index, float mass, const float g[3])
{
  /* force = mass * acceleration (in this case: gravity) */
  float f[3];
  world_to_root_v3(data, index, f, g);
  mul_v3_fl(f, mass);
 
  add_v3_v3(data->F[index], f);
}
 
void SIM_mass_spring_force_drag(Implicit_Data *data, float drag)
{
  int i, numverts = data->M[0].vcount;
  for (i = 0; i < numverts; i++) {
    float tmp[3][3];
 
    /* NOTE: Uses root space velocity, no need to transform. */
    madd_v3_v3fl(data->F[i], data->V[i], -drag);
 
    copy_m3_m3(tmp, I);
    mul_m3_fl(tmp, -drag);
    add_m3_m3m3(data->dFdV[i].m, data->dFdV[i].m, tmp);
  }
}
 
void SIM_mass_spring_force_extern(
    Implicit_Data *data, int i, const float f[3], float dfdx[3][3], float dfdv[3][3])
{
  float tf[3], tdfdx[3][3], tdfdv[3][3];
  world_to_root_v3(data, i, tf, f);
  world_to_root_m3(data, i, tdfdx, dfdx);
  world_to_root_m3(data, i, tdfdv, dfdv);
 
  add_v3_v3(data->F[i], tf);
  add_m3_m3m3(data->dFdX[i].m, data->dFdX[i].m, tdfdx);
  add_m3_m3m3(data->dFdV[i].m, data->dFdV[i].m, tdfdv);
}
 
static float calc_nor_area_tri(float nor[3],
                               const float v1[3],
                               const float v2[3],
                               const float v3[3])
{
  float n1[3], n2[3];
 
  sub_v3_v3v3(n1, v1, v2);
  sub_v3_v3v3(n2, v2, v3);
 
  cross_v3_v3v3(nor, n1, n2);
  return normalize_v3(nor) / 2.0f;
}
 
void SIM_mass_spring_force_face_wind(
    Implicit_Data *data, int v1, int v2, int v3, const float (*winvec)[3])
{
  /* XXX does not support force jacobians yet,
   * since the effector system does not provide them either. */
 
  const float effector_scale = 0.02f;
  const int vs[3] = {v1, v2, v3};
  float win[3], nor[3], area;
  float factor, base_force;
  float force[3];
 
  /* calculate face normal and area */
  area = calc_nor_area_tri(nor, data->X[v1], data->X[v2], data->X[v3]);
  /* The force is calculated and split up evenly for each of the three face verts */
  factor = effector_scale * area / 3.0f;
 
  /* Calculate wind pressure at each vertex by projecting the wind field on the normal. */
  for (int i = 0; i < 3; i++) {
    world_to_root_v3(data, vs[i], win, winvec[vs[i]]);
 
    force[i] = dot_v3v3(win, nor);
  }
 
  /* Compute per-vertex force values from local pressures.
   * From integrating the pressure over the triangle and deriving
   * equivalent vertex forces, it follows that:
   *
   * force[idx] = (sum(pressure) + pressure[idx]) * area / 12
   *
   * Effectively, 1/4 of the pressure acts just on its vertex,
   * while 3/4 is split evenly over all three.
   */
  mul_v3_fl(force, factor / 4.0f);
 
  base_force = force[0] + force[1] + force[2];
 
  /* add pressure to each of the face verts */
  madd_v3_v3fl(data->F[v1], nor, base_force + force[0]);
  madd_v3_v3fl(data->F[v2], nor, base_force + force[1]);
  madd_v3_v3fl(data->F[v3], nor, base_force + force[2]);
}
 
void SIM_mass_spring_force_face_extern(
    Implicit_Data *data, int v1, int v2, int v3, const float (*forcevec)[3])
{
  const float effector_scale = 0.02f;
  const int vs[3] = {v1, v2, v3};
  float nor[3], area;
  float factor, base_force[3];
  float force[3][3];
 
  /* calculate face normal and area */
  area = calc_nor_area_tri(nor, data->X[v1], data->X[v2], data->X[v3]);
  /* The force is calculated and split up evenly for each of the three face verts */
  factor = effector_scale * area / 3.0f;
 
  /* Compute common and per-vertex force vectors from the original inputs. */
  zero_v3(base_force);
 
  for (int i = 0; i < 3; i++) {
    world_to_root_v3(data, vs[i], force[i], forcevec[vs[i]]);
 
    mul_v3_fl(force[i], factor / 4.0f);
    add_v3_v3(base_force, force[i]);
  }
 
  /* Apply the common and vertex components to all vertices. */
  for (int i = 0; i < 3; i++) {
    add_v3_v3(force[i], base_force);
    add_v3_v3(data->F[vs[i]], force[i]);
  }
}
 
float SIM_tri_tetra_volume_signed_6x(Implicit_Data *data, int v1, int v2, int v3)
{
  /* The result will be 6x the volume */
  return volume_tri_tetrahedron_signed_v3_6x(data->X[v1], data->X[v2], data->X[v3]);
}
 
float SIM_tri_area(Implicit_Data *data, int v1, int v2, int v3)
{
  float nor[3];
 
  return calc_nor_area_tri(nor, data->X[v1], data->X[v2], data->X[v3]);
}
 
void SIM_mass_spring_force_pressure(Implicit_Data *data,
                                    int v1,
                                    int v2,
                                    int v3,
                                    float common_pressure,
                                    const float *vertex_pressure,
                                    const float weights[3])
{
  float nor[3], area;
  float factor, base_force;
  float force[3];
 
  /* calculate face normal and area */
  area = calc_nor_area_tri(nor, data->X[v1], data->X[v2], data->X[v3]);
  /* The force is calculated and split up evenly for each of the three face verts */
  factor = area / 3.0f;
  base_force = common_pressure * factor;
 
  /* Compute per-vertex force values from local pressures.
   * From integrating the pressure over the triangle and deriving
   * equivalent vertex forces, it follows that:
   *
   * force[idx] = (sum(pressure) + pressure[idx]) * area / 12
   *
   * Effectively, 1/4 of the pressure acts just on its vertex,
   * while 3/4 is split evenly over all three.
   */
  if (vertex_pressure) {
    copy_v3_fl3(force, vertex_pressure[v1], vertex_pressure[v2], vertex_pressure[v3]);
    mul_v3_fl(force, factor / 4.0f);
 
    base_force += force[0] + force[1] + force[2];
  }
  else {
    zero_v3(force);
  }
 
  /* add pressure to each of the face verts */
  madd_v3_v3fl(data->F[v1], nor, (base_force + force[0]) * weights[0]);
  madd_v3_v3fl(data->F[v2], nor, (base_force + force[1]) * weights[1]);
  madd_v3_v3fl(data->F[v3], nor, (base_force + force[2]) * weights[2]);
}
 
static void edge_wind_vertex(const float dir[3],
                             float length,
                             float radius,
                             const float wind[3],
                             float f[3],
                             float /*dfdx*/[3][3],
                             float /*dfdv*/[3][3])
{
  const float density = 0.01f; /* XXX arbitrary value, corresponds to effect of air density */
  float cos_alpha, sin_alpha, cross_section;
  float windlen = len_v3(wind);
 
  if (windlen == 0.0f) {
    zero_v3(f);
    return;
  }
 
  /* angle of wind direction to edge */
  cos_alpha = dot_v3v3(wind, dir) / windlen;
  sin_alpha = sqrtf(1.0f - cos_alpha * cos_alpha);
  cross_section = radius * (float(M_PI) * radius * sin_alpha + length * cos_alpha);
 
  mul_v3_v3fl(f, wind, density * cross_section);
}
 
void SIM_mass_spring_force_edge_wind(
    Implicit_Data *data, int v1, int v2, float radius1, float radius2, const float (*winvec)[3])
{
  float win[3], dir[3], length;
  float f[3], dfdx[3][3], dfdv[3][3];
 
  sub_v3_v3v3(dir, data->X[v1], data->X[v2]);
  length = normalize_v3(dir);
 
  world_to_root_v3(data, v1, win, winvec[v1]);
  edge_wind_vertex(dir, length, radius1, win, f, dfdx, dfdv);
  add_v3_v3(data->F[v1], f);
 
  world_to_root_v3(data, v2, win, winvec[v2]);
  edge_wind_vertex(dir, length, radius2, win, f, dfdx, dfdv);
  add_v3_v3(data->F[v2], f);
}
 
void SIM_mass_spring_force_vertex_wind(Implicit_Data *data,
                                       int v,
                                       float /*radius*/,
                                       const float (*winvec)[3])
{
  const float density = 0.01f; /* XXX arbitrary value, corresponds to effect of air density */
 
  float wind[3];
  float f[3];
 
  world_to_root_v3(data, v, wind, winvec[v]);
  mul_v3_v3fl(f, wind, density);
  add_v3_v3(data->F[v], f);
}
 
BLI_INLINE void dfdx_spring(float to[3][3], const float dir[3], float length, float L, float k)
{
  /* dir is unit length direction, rest is spring's restlength, k is spring constant. */
  // return  ( (I-outerprod(dir, dir))*Min(1.0f, rest/length) - I) * -k;
  outerproduct(to, dir, dir);
  sub_m3_m3m3(to, I, to);
 
  mul_m3_fl(to, (L / length));
  sub_m3_m3m3(to, to, I);
  mul_m3_fl(to, k);
}
 
/* unused */
#  if 0
BLI_INLINE void dfdx_damp(float to[3][3],
                          const float dir[3],
                          float length,
                          const float vel[3],
                          float rest,
                          float damping)
{
  /* Inner spring damping `vel` is the relative velocity of the endpoints. */
  // return (I - outerprod(dir, dir)) * (-damping * -(dot(dir, vel) / Max(length, rest)));
  mul_fvectorT_fvector(to, dir, dir);
  sub_fmatrix_fmatrix(to, I, to);
  mul_fmatrix_S(to, (-damping * -(dot_v3v3(dir, vel) / std::max(length, rest))));
}
#  endif
 
BLI_INLINE void dfdv_damp(float to[3][3], const float dir[3], float damping)
{
  /* Derivative of force with regards to velocity. */
  outerproduct(to, dir, dir);
  mul_m3_fl(to, -damping);
}
 
BLI_INLINE float fb(float length, float L)
{
  float x = length / L;
  float xx = x * x;
  float xxx = xx * x;
  float xxxx = xxx * x;
  return (-11.541f * xxxx + 34.193f * xxx - 39.083f * xx + 23.116f * x - 9.713f);
}
 
BLI_INLINE float fbderiv(float length, float L)
{
  float x = length / L;
  float xx = x * x;
  float xxx = xx * x;
  return (-46.164f * xxx + 102.579f * xx - 78.166f * x + 23.116f);
}
 
BLI_INLINE float fbstar(float length, float L, float kb, float cb)
{
  float tempfb_fl = kb * fb(length, L);
  float fbstar_fl = cb * (length - L);
 
  if (tempfb_fl < fbstar_fl) {
    return fbstar_fl;
  }
 
  return tempfb_fl;
}
 
/* Function to calculate bending spring force (taken from Choi & Co). */
BLI_INLINE float fbstar_jacobi(float length, float L, float kb, float cb)
{
  float tempfb_fl = kb * fb(length, L);
  float fbstar_fl = cb * (length - L);
 
  if (tempfb_fl < fbstar_fl) {
    return -cb;
  }
 
  return -kb * fbderiv(length, L);
}
 
/* calculate elongation */
BLI_INLINE bool spring_length(Implicit_Data *data,
                              int i,
                              int j,
                              float r_extent[3],
                              float r_dir[3],
                              float *r_length,
                              float r_vel[3])
{
  sub_v3_v3v3(r_extent, data->X[j], data->X[i]);
  sub_v3_v3v3(r_vel, data->V[j], data->V[i]);
  *r_length = len_v3(r_extent);
 
  if (*r_length > ALMOST_ZERO) {
#  if 0
    if (length > L) {
      if ((clmd->sim_parms->flags & CSIMSETT_FLAG_TEARING_ENABLED) &&
          (((length - L) * 100.0f / L) > clmd->sim_parms->maxspringlen))
      {
        /* cut spring! */
        s->flags |= CSPRING_FLAG_DEACTIVATE;
        return false;
      }
    }
#  endif
    mul_v3_v3fl(r_dir, r_extent, 1.0f / (*r_length));
  }
  else {
    zero_v3(r_dir);
  }
 
  return true;
}
 
BLI_INLINE void apply_spring(Implicit_Data *data,
                             int i,
                             int j,
                             const float f[3],
                             const float dfdx[3][3],
                             const float dfdv[3][3])
{
  int block_ij = SIM_mass_spring_add_block(data, i, j);
 
  add_v3_v3(data->F[i], f);
  sub_v3_v3(data->F[j], f);
 
  add_m3_m3m3(data->dFdX[i].m, data->dFdX[i].m, dfdx);
  add_m3_m3m3(data->dFdX[j].m, data->dFdX[j].m, dfdx);
  sub_m3_m3m3(data->dFdX[block_ij].m, data->dFdX[block_ij].m, dfdx);
 
  add_m3_m3m3(data->dFdV[i].m, data->dFdV[i].m, dfdv);
  add_m3_m3m3(data->dFdV[j].m, data->dFdV[j].m, dfdv);
  sub_m3_m3m3(data->dFdV[block_ij].m, data->dFdV[block_ij].m, dfdv);
}
 
bool SIM_mass_spring_force_spring_linear(Implicit_Data *data,
                                         int i,
                                         int j,
                                         float restlen,
                                         float stiffness_tension,
                                         float damping_tension,
                                         float stiffness_compression,
                                         float damping_compression,
                                         bool resist_compress,
                                         bool new_compress,
                                         float clamp_force)
{
  float extent[3], length, dir[3], vel[3];
  float f[3], dfdx[3][3], dfdv[3][3];
  float damping = 0;
 
  /* calculate elongation */
  spring_length(data, i, j, extent, dir, &length, vel);
 
  /* This code computes not only the force, but also its derivative.
   * Zero derivative effectively disables the spring for the implicit solver.
   * Thus length > restlen makes cloth unconstrained at the start of simulation. */
  if ((length >= restlen && length > 0) || resist_compress) {
    float stretch_force;
 
    damping = damping_tension;
 
    stretch_force = stiffness_tension * (length - restlen);
    if (clamp_force > 0.0f && stretch_force > clamp_force) {
      stretch_force = clamp_force;
    }
    mul_v3_v3fl(f, dir, stretch_force);
 
    dfdx_spring(dfdx, dir, length, restlen, stiffness_tension);
  }
  else if (new_compress) {
    /* This is based on the Choi and Ko bending model,
     * which works surprisingly well for compression. */
    float kb = stiffness_compression;
    float cb = kb; /* cb equal to kb seems to work, but a factor can be added if necessary */
 
    damping = damping_compression;
 
    mul_v3_v3fl(f, dir, fbstar(length, restlen, kb, cb));
 
    outerproduct(dfdx, dir, dir);
    mul_m3_fl(dfdx, fbstar_jacobi(length, restlen, kb, cb));
  }
  else {
    return false;
  }
 
  madd_v3_v3fl(f, dir, damping * dot_v3v3(vel, dir));
  dfdv_damp(dfdv, dir, damping);
 
  apply_spring(data, i, j, f, dfdx, dfdv);
 
  return true;
}
 
bool SIM_mass_spring_force_spring_bending(
    Implicit_Data *data, int i, int j, float restlen, float kb, float cb)
{
  /* See "Stable but Responsive Cloth" (Choi, Ko 2005). */
 
  float extent[3], length, dir[3], vel[3];
 
  /* calculate elongation */
  spring_length(data, i, j, extent, dir, &length, vel);
 
  if (length < restlen) {
    float f[3], dfdx[3][3], dfdv[3][3];
 
    mul_v3_v3fl(f, dir, fbstar(length, restlen, kb, cb));
 
    outerproduct(dfdx, dir, dir);
    mul_m3_fl(dfdx, fbstar_jacobi(length, restlen, kb, cb));
 
    /* XXX damping not supported */
    zero_m3(dfdv);
 
    apply_spring(data, i, j, f, dfdx, dfdv);
 
    return true;
  }
 
  return false;
}
 
BLI_INLINE void poly_avg(lfVector *data, const int *inds, int len, float r_avg[3])
{
  float fact = 1.0f / float(len);
 
  zero_v3(r_avg);
 
  for (int i = 0; i < len; i++) {
    madd_v3_v3fl(r_avg, data[inds[i]], fact);
  }
}
 
BLI_INLINE void poly_norm(lfVector *data, int i, int j, int *inds, int len, float r_dir[3])
{
  float mid[3];
 
  poly_avg(data, inds, len, mid);
 
  normal_tri_v3(r_dir, data[i], data[j], mid);
}
 
BLI_INLINE void edge_avg(lfVector *data, int i, int j, float r_avg[3])
{
  r_avg[0] = (data[i][0] + data[j][0]) * 0.5f;
  r_avg[1] = (data[i][1] + data[j][1]) * 0.5f;
  r_avg[2] = (data[i][2] + data[j][2]) * 0.5f;
}
 
BLI_INLINE void edge_norm(lfVector *data, int i, int j, float r_dir[3])
{
  sub_v3_v3v3(r_dir, data[i], data[j]);
  normalize_v3(r_dir);
}
 
BLI_INLINE float bend_angle(const float dir_a[3], const float dir_b[3], const float dir_e[3])
{
  float cos, sin;
  float tmp[3];
 
  cos = dot_v3v3(dir_a, dir_b);
 
  cross_v3_v3v3(tmp, dir_a, dir_b);
  sin = dot_v3v3(tmp, dir_e);
 
  return atan2f(sin, cos);
}
 
BLI_INLINE void spring_angle(Implicit_Data *data,
                             int i,
                             int j,
                             int *i_a,
                             int *i_b,
                             int len_a,
                             int len_b,
                             float r_dir_a[3],
                             float r_dir_b[3],
                             float *r_angle,
                             float r_vel_a[3],
                             float r_vel_b[3])
{
  float dir_e[3], vel_e[3];
 
  poly_norm(data->X, j, i, i_a, len_a, r_dir_a);
  poly_norm(data->X, i, j, i_b, len_b, r_dir_b);
 
  edge_norm(data->X, i, j, dir_e);
 
  *r_angle = bend_angle(r_dir_a, r_dir_b, dir_e);
 
  poly_avg(data->V, i_a, len_a, r_vel_a);
  poly_avg(data->V, i_b, len_b, r_vel_b);
 
  edge_avg(data->V, i, j, vel_e);
 
  sub_v3_v3(r_vel_a, vel_e);
  sub_v3_v3(r_vel_b, vel_e);
}
 
bool SIM_mass_spring_force_spring_angular(Implicit_Data *data,
                                          int i,
                                          int j,
                                          int *i_a,
                                          int *i_b,
                                          int len_a,
                                          int len_b,
                                          float restang,
                                          float stiffness,
                                          float damping)
{
  float angle, dir_a[3], dir_b[3], vel_a[3], vel_b[3];
  float f_a[3], f_b[3], f_e[3];
  float force;
  int x;
 
  spring_angle(data, i, j, i_a, i_b, len_a, len_b, dir_a, dir_b, &angle, vel_a, vel_b);
 
  /* spring force */
  force = stiffness * (angle - restang);
 
  /* damping force */
  force += -damping * (dot_v3v3(vel_a, dir_a) + dot_v3v3(vel_b, dir_b));
 
  mul_v3_v3fl(f_a, dir_a, force / len_a);
  mul_v3_v3fl(f_b, dir_b, force / len_b);
 
  for (x = 0; x < len_a; x++) {
    add_v3_v3(data->F[i_a[x]], f_a);
  }
 
  for (x = 0; x < len_b; x++) {
    add_v3_v3(data->F[i_b[x]], f_b);
  }
 
  mul_v3_v3fl(f_a, dir_a, force * 0.5f);
  mul_v3_v3fl(f_b, dir_b, force * 0.5f);
 
  add_v3_v3v3(f_e, f_a, f_b);
 
  sub_v3_v3(data->F[i], f_e);
  sub_v3_v3(data->F[j], f_e);
 
  return true;
}
 
/* Jacobian of a direction vector.
 * Basically the part of the differential orthogonal to the direction,
 * inversely proportional to the length of the edge.
 *
 * dD_ij/dx_i = -dD_ij/dx_j = (D_ij * D_ij^T - I) / len_ij
 */
BLI_INLINE void spring_grad_dir(
    Implicit_Data *data, int i, int j, float edge[3], float dir[3], float grad_dir[3][3])
{
  float length;
 
  sub_v3_v3v3(edge, data->X[j], data->X[i]);
  length = normalize_v3_v3(dir, edge);
 
  if (length > ALMOST_ZERO) {
    outerproduct(grad_dir, dir, dir);
    sub_m3_m3m3(grad_dir, I, grad_dir);
    mul_m3_fl(grad_dir, 1.0f / length);
  }
  else {
    zero_m3(grad_dir);
  }
}
 
BLI_INLINE void spring_hairbend_forces(Implicit_Data *data,
                                       int i,
                                       int j,
                                       int k,
                                       const float goal[3],
                                       float stiffness,
                                       float damping,
                                       int q,
                                       const float dx[3],
                                       const float dv[3],
                                       float r_f[3])
{
  float edge_ij[3], dir_ij[3];
  float edge_jk[3], dir_jk[3];
  float vel_ij[3], vel_jk[3], vel_ortho[3];
  float f_bend[3], f_damp[3];
  float fk[3];
  float dist[3];
 
  zero_v3(fk);
 
  sub_v3_v3v3(edge_ij, data->X[j], data->X[i]);
  if (q == i) {
    sub_v3_v3(edge_ij, dx);
  }
  if (q == j) {
    add_v3_v3(edge_ij, dx);
  }
  normalize_v3_v3(dir_ij, edge_ij);
 
  sub_v3_v3v3(edge_jk, data->X[k], data->X[j]);
  if (q == j) {
    sub_v3_v3(edge_jk, dx);
  }
  if (q == k) {
    add_v3_v3(edge_jk, dx);
  }
  normalize_v3_v3(dir_jk, edge_jk);
 
  sub_v3_v3v3(vel_ij, data->V[j], data->V[i]);
  if (q == i) {
    sub_v3_v3(vel_ij, dv);
  }
  if (q == j) {
    add_v3_v3(vel_ij, dv);
  }
 
  sub_v3_v3v3(vel_jk, data->V[k], data->V[j]);
  if (q == j) {
    sub_v3_v3(vel_jk, dv);
  }
  if (q == k) {
    add_v3_v3(vel_jk, dv);
  }
 
  /* bending force */
  sub_v3_v3v3(dist, goal, edge_jk);
  mul_v3_v3fl(f_bend, dist, stiffness);
 
  add_v3_v3(fk, f_bend);
 
  /* damping force */
  madd_v3_v3v3fl(vel_ortho, vel_jk, dir_jk, -dot_v3v3(vel_jk, dir_jk));
  mul_v3_v3fl(f_damp, vel_ortho, damping);
 
  sub_v3_v3(fk, f_damp);
 
  copy_v3_v3(r_f, fk);
}
 
/* Finite Differences method for estimating the jacobian of the force */
BLI_INLINE void spring_hairbend_estimate_dfdx(Implicit_Data *data,
                                              int i,
                                              int j,
                                              int k,
                                              const float goal[3],
                                              float stiffness,
                                              float damping,
                                              int q,
                                              float dfdx[3][3])
{
  const float delta = 0.00001f; /* TODO: find a good heuristic for this. */
  float dvec_null[3][3], dvec_pos[3][3], dvec_neg[3][3];
  float f[3];
  int a, b;
 
  zero_m3(dvec_null);
  unit_m3(dvec_pos);
  mul_m3_fl(dvec_pos, delta * 0.5f);
  copy_m3_m3(dvec_neg, dvec_pos);
  negate_m3(dvec_neg);
 
  /* XXX TODO: offset targets to account for position dependency. */
 
  for (a = 0; a < 3; a++) {
    spring_hairbend_forces(
        data, i, j, k, goal, stiffness, damping, q, dvec_pos[a], dvec_null[a], f);
    copy_v3_v3(dfdx[a], f);
 
    spring_hairbend_forces(
        data, i, j, k, goal, stiffness, damping, q, dvec_neg[a], dvec_null[a], f);
    sub_v3_v3(dfdx[a], f);
 
    for (b = 0; b < 3; b++) {
      dfdx[a][b] /= delta;
    }
  }
}
 
/* Finite Differences method for estimating the jacobian of the force */
BLI_INLINE void spring_hairbend_estimate_dfdv(Implicit_Data *data,
                                              int i,
                                              int j,
                                              int k,
                                              const float goal[3],
                                              float stiffness,
                                              float damping,
                                              int q,
                                              float dfdv[3][3])
{
  const float delta = 0.00001f; /* TODO: find a good heuristic for this. */
  float dvec_null[3][3], dvec_pos[3][3], dvec_neg[3][3];
  float f[3];
  int a, b;
 
  zero_m3(dvec_null);
  unit_m3(dvec_pos);
  mul_m3_fl(dvec_pos, delta * 0.5f);
  copy_m3_m3(dvec_neg, dvec_pos);
  negate_m3(dvec_neg);
 
  /* XXX TODO: offset targets to account for position dependency. */
 
  for (a = 0; a < 3; a++) {
    spring_hairbend_forces(
        data, i, j, k, goal, stiffness, damping, q, dvec_null[a], dvec_pos[a], f);
    copy_v3_v3(dfdv[a], f);
 
    spring_hairbend_forces(
        data, i, j, k, goal, stiffness, damping, q, dvec_null[a], dvec_neg[a], f);
    sub_v3_v3(dfdv[a], f);
 
    for (b = 0; b < 3; b++) {
      dfdv[a][b] /= delta;
    }
  }
}
 
bool SIM_mass_spring_force_spring_bending_hair(Implicit_Data *data,
                                               int i,
                                               int j,
                                               int k,
                                               const float target[3],
                                               float stiffness,
                                               float damping)
{
  /* Angular springs roughly based on the bending model proposed by Baraff and Witkin in
   * "Large Steps in Cloth Simulation". */
 
  float goal[3];
  float fj[3], fk[3];
  float dfj_dxi[3][3], dfj_dxj[3][3], dfk_dxi[3][3], dfk_dxj[3][3], dfk_dxk[3][3];
  float dfj_dvi[3][3], dfj_dvj[3][3], dfk_dvi[3][3], dfk_dvj[3][3], dfk_dvk[3][3];
 
  const float vecnull[3] = {0.0f, 0.0f, 0.0f};
 
  int block_ij = SIM_mass_spring_add_block(data, i, j);
  int block_jk = SIM_mass_spring_add_block(data, j, k);
  int block_ik = SIM_mass_spring_add_block(data, i, k);
 
  world_to_root_v3(data, j, goal, target);
 
  spring_hairbend_forces(data, i, j, k, goal, stiffness, damping, k, vecnull, vecnull, fk);
  negate_v3_v3(fj, fk); /* Counter-force. */
 
  spring_hairbend_estimate_dfdx(data, i, j, k, goal, stiffness, damping, i, dfk_dxi);
  spring_hairbend_estimate_dfdx(data, i, j, k, goal, stiffness, damping, j, dfk_dxj);
  spring_hairbend_estimate_dfdx(data, i, j, k, goal, stiffness, damping, k, dfk_dxk);
  copy_m3_m3(dfj_dxi, dfk_dxi);
  negate_m3(dfj_dxi);
  copy_m3_m3(dfj_dxj, dfk_dxj);
  negate_m3(dfj_dxj);
 
  spring_hairbend_estimate_dfdv(data, i, j, k, goal, stiffness, damping, i, dfk_dvi);
  spring_hairbend_estimate_dfdv(data, i, j, k, goal, stiffness, damping, j, dfk_dvj);
  spring_hairbend_estimate_dfdv(data, i, j, k, goal, stiffness, damping, k, dfk_dvk);
  copy_m3_m3(dfj_dvi, dfk_dvi);
  negate_m3(dfj_dvi);
  copy_m3_m3(dfj_dvj, dfk_dvj);
  negate_m3(dfj_dvj);
 
  /* add forces and jacobians to the solver data */
 
  add_v3_v3(data->F[j], fj);
  add_v3_v3(data->F[k], fk);
 
  add_m3_m3m3(data->dFdX[j].m, data->dFdX[j].m, dfj_dxj);
  add_m3_m3m3(data->dFdX[k].m, data->dFdX[k].m, dfk_dxk);
 
  add_m3_m3m3(data->dFdX[block_ij].m, data->dFdX[block_ij].m, dfj_dxi);
  add_m3_m3m3(data->dFdX[block_jk].m, data->dFdX[block_jk].m, dfk_dxj);
  add_m3_m3m3(data->dFdX[block_ik].m, data->dFdX[block_ik].m, dfk_dxi);
 
  add_m3_m3m3(data->dFdV[j].m, data->dFdV[j].m, dfj_dvj);
  add_m3_m3m3(data->dFdV[k].m, data->dFdV[k].m, dfk_dvk);
 
  add_m3_m3m3(data->dFdV[block_ij].m, data->dFdV[block_ij].m, dfj_dvi);
  add_m3_m3m3(data->dFdV[block_jk].m, data->dFdV[block_jk].m, dfk_dvj);
  add_m3_m3m3(data->dFdV[block_ik].m, data->dFdV[block_ik].m, dfk_dvi);
 
/* XXX analytical calculation of derivatives below is incorrect.
 * This proved to be difficult, but for now just using the finite difference method for
 * estimating the jacobians should be sufficient.
 */
#  if 0
  float edge_ij[3], dir_ij[3], grad_dir_ij[3][3];
  float edge_jk[3], dir_jk[3], grad_dir_jk[3][3];
  float dist[3], vel_jk[3], vel_jk_ortho[3], projvel[3];
  float target[3];
  float tmp[3][3];
  float fi[3], fj[3], fk[3];
  float dfi_dxi[3][3], dfj_dxi[3][3], dfj_dxj[3][3], dfk_dxi[3][3], dfk_dxj[3][3], dfk_dxk[3][3];
  float dfdvi[3][3];
 
  /* TESTING */
  damping = 0.0f;
 
  zero_v3(fi);
  zero_v3(fj);
  zero_v3(fk);
  zero_m3(dfi_dxi);
  zero_m3(dfj_dxi);
  zero_m3(dfk_dxi);
  zero_m3(dfk_dxj);
  zero_m3(dfk_dxk);
 
  /* jacobian of direction vectors */
  spring_grad_dir(data, i, j, edge_ij, dir_ij, grad_dir_ij);
  spring_grad_dir(data, j, k, edge_jk, dir_jk, grad_dir_jk);
 
  sub_v3_v3v3(vel_jk, data->V[k], data->V[j]);
 
  /* bending force */
  mul_v3_v3fl(target, dir_ij, restlen);
  sub_v3_v3v3(dist, target, edge_jk);
  mul_v3_v3fl(fk, dist, stiffness);
 
  /* damping force */
  madd_v3_v3v3fl(vel_jk_ortho, vel_jk, dir_jk, -dot_v3v3(vel_jk, dir_jk));
  madd_v3_v3fl(fk, vel_jk_ortho, damping);
 
  /* XXX this only holds true as long as we assume straight rest shape!
   * eventually will become a bit more involved since the opposite segment
   * gets its own target, under condition of having equal torque on both sides.
   */
  copy_v3_v3(fi, fk);
 
  /* counterforce on the middle point */
  sub_v3_v3(fj, fi);
  sub_v3_v3(fj, fk);
 
  /* === derivatives === */
 
  madd_m3_m3fl(dfk_dxi, grad_dir_ij, stiffness * restlen);
 
  madd_m3_m3fl(dfk_dxj, grad_dir_ij, -stiffness * restlen);
  madd_m3_m3fl(dfk_dxj, I, stiffness);
 
  madd_m3_m3fl(dfk_dxk, I, -stiffness);
 
  copy_m3_m3(dfi_dxi, dfk_dxk);
  negate_m3(dfi_dxi);
 
  /* dfj_dfi == dfi_dfj due to symmetry,
   * dfi_dfj == dfk_dfj due to fi == fk
   * XXX see comment above on future bent rest shapes
   */
  copy_m3_m3(dfj_dxi, dfk_dxj);
 
  /* dfj_dxj == -(dfi_dxj + dfk_dxj) due to fj == -(fi + fk) */
  sub_m3_m3m3(dfj_dxj, dfj_dxj, dfj_dxi);
  sub_m3_m3m3(dfj_dxj, dfj_dxj, dfk_dxj);
 
  /* add forces and jacobians to the solver data */
  add_v3_v3(data->F[i], fi);
  add_v3_v3(data->F[j], fj);
  add_v3_v3(data->F[k], fk);
 
  add_m3_m3m3(data->dFdX[i].m, data->dFdX[i].m, dfi_dxi);
  add_m3_m3m3(data->dFdX[j].m, data->dFdX[j].m, dfj_dxj);
  add_m3_m3m3(data->dFdX[k].m, data->dFdX[k].m, dfk_dxk);
 
  add_m3_m3m3(data->dFdX[block_ij].m, data->dFdX[block_ij].m, dfj_dxi);
  add_m3_m3m3(data->dFdX[block_jk].m, data->dFdX[block_jk].m, dfk_dxj);
  add_m3_m3m3(data->dFdX[block_ik].m, data->dFdX[block_ik].m, dfk_dxi);
#  endif
 
  return true;
}
 
bool SIM_mass_spring_force_spring_goal(Implicit_Data *data,
                                       int i,
                                       const float goal_x[3],
                                       const float goal_v[3],
                                       float stiffness,
                                       float damping)
{
  float root_goal_x[3], root_goal_v[3], extent[3], length, dir[3], vel[3];
  float f[3], dfdx[3][3], dfdv[3][3];
 
  /* goal is in world space */
  world_to_root_v3(data, i, root_goal_x, goal_x);
  world_to_root_v3(data, i, root_goal_v, goal_v);
 
  sub_v3_v3v3(extent, root_goal_x, data->X[i]);
  sub_v3_v3v3(vel, root_goal_v, data->V[i]);
  length = normalize_v3_v3(dir, extent);
 
  if (length > ALMOST_ZERO) {
    mul_v3_v3fl(f, dir, stiffness * length);
 
    /* Ascher & Boxman, p.21: Damping only during elongation
     * something wrong with it. */
    madd_v3_v3fl(f, dir, damping * dot_v3v3(vel, dir));
 
    dfdx_spring(dfdx, dir, length, 0.0f, stiffness);
    dfdv_damp(dfdv, dir, damping);
 
    add_v3_v3(data->F[i], f);
    add_m3_m3m3(data->dFdX[i].m, data->dFdX[i].m, dfdx);
    add_m3_m3m3(data->dFdV[i].m, data->dFdV[i].m, dfdv);
 
    return true;
  }
 
  return false;
}
 
#endif /* IMPLICIT_SOLVER_BLENDER */