blender-archive/source/blender/physics/intern/implicit_eigen.cpp

/*
 * This program is free software; you can redistribute it and/or
 * modify it under the terms of the GNU General Public License
 * as published by the Free Software Foundation; either version 2
 * of the License, or (at your option) any later version.
 *
 * This program 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 General Public License for more details.
 *
 * You should have received a copy of the GNU General Public License
 * along with this program; if not, write to the Free Software Foundation,
 * Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
 *
 * The Original Code is Copyright (C) Blender Foundation
 * All rights reserved.
 */

/** \file
 * \ingroup bph
 */

#include "implicit.h"

#ifdef IMPLICIT_SOLVER_EIGEN

//#define USE_EIGEN_CORE
#define USE_EIGEN_CONSTRAINED_CG

#ifdef __GNUC__
#  pragma GCC diagnostic push
/* XXX suppress verbose warnings in eigen */
//#  pragma GCC diagnostic ignored "-Wlogical-op"
#endif

#ifndef IMPLICIT_ENABLE_EIGEN_DEBUG
#ifdef NDEBUG
#define IMPLICIT_NDEBUG
#endif
#define NDEBUG
#endif

#include <Eigen/Sparse>
#include <Eigen/src/Core/util/DisableStupidWarnings.h>

#ifdef USE_EIGEN_CONSTRAINED_CG
#include <intern/ConstrainedConjugateGradient.h>
#endif

#ifndef IMPLICIT_ENABLE_EIGEN_DEBUG
#ifndef IMPLICIT_NDEBUG
#undef NDEBUG
#else
#undef IMPLICIT_NDEBUG
#endif
#endif

#ifdef __GNUC__
#  pragma GCC diagnostic pop
#endif

#include "MEM_guardedalloc.h"

extern "C" {
#include "DNA_scene_types.h"
#include "DNA_object_types.h"
#include "DNA_object_force_types.h"
#include "DNA_meshdata_types.h"
#include "DNA_texture_types.h"

#include "BLI_math.h"
#include "BLI_linklist.h"
#include "BLI_utildefines.h"

#include "BKE_cloth.h"
#include "BKE_collision.h"
#include "BKE_effect.h"
#include "BKE_global.h"

#include "BPH_mass_spring.h"
}

typedef float Scalar;

static float I[3][3] = {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}};

/* slightly extended Eigen vector class
 * with conversion to/from plain C float array
 */
class fVector : public Eigen::Vector3f {
public:
	typedef float *ctype;

	fVector()
	{
	}

	fVector(const ctype &v)
	{
		for (int k = 0; k < 3; ++k)
			coeffRef(k) = v[k];
	}

	fVector& operator =(const ctype &v)
	{
		for (int k = 0; k < 3; ++k)
			coeffRef(k) = v[k];
		return *this;
	}

	operator ctype()
	{
		return data();
	}
};

/* slightly extended Eigen matrix class
 * with conversion to/from plain C float array
 */
class fMatrix : public Eigen::Matrix3f {
public:
	typedef float (*ctype)[3];

	fMatrix()
	{
	}

	fMatrix(const ctype &v)
	{
		for (int k = 0; k < 3; ++k)
			for (int l = 0; l < 3; ++l)
				coeffRef(l, k) = v[k][l];
	}

	fMatrix& operator =(const ctype &v)
	{
		for (int k = 0; k < 3; ++k)
			for (int l = 0; l < 3; ++l)
				coeffRef(l, k) = v[k][l];
		return *this;
	}

	operator ctype()
	{
		return (ctype)data();
	}
};

/* Extension of dense Eigen vectors,
 * providing 3-float block access for blenlib math functions
 */
class lVector : public Eigen::VectorXf {
public:
	typedef Eigen::VectorXf base_t;

	lVector()
	{
	}

	template <typename T>
	lVector& operator =(T rhs)
	{
		base_t::operator=(rhs);
		return *this;
	}

	float *v3(int vertex)
	{
		return &coeffRef(3 * vertex);
	}

	const float *v3(int vertex) const
	{
		return &coeffRef(3 * vertex);
	}
};

typedef Eigen::Triplet<Scalar> Triplet;
typedef std::vector<Triplet> TripletList;

typedef Eigen::SparseMatrix<Scalar> lMatrix;

/* Constructor type that provides more convenient handling of Eigen triplets
 * for efficient construction of sparse 3x3 block matrices.
 * This should be used for building lMatrix instead of writing to such lMatrix directly (which is very inefficient).
 * After all elements have been defined using the set() method, the actual matrix can be filled using construct().
 */
struct lMatrixCtor {
	lMatrixCtor()
	{
	}

	void reset()
	{
		m_trips.clear();
	}

	void reserve(int numverts)
	{
		/* reserve for diagonal entries */
		m_trips.reserve(numverts * 9);
	}

	void add(int i, int j, const fMatrix &m)
	{
		i *= 3;
		j *= 3;
		for (int k = 0; k < 3; ++k)
			for (int l = 0; l < 3; ++l)
				m_trips.push_back(Triplet(i + k, j + l, m.coeff(l, k)));
	}

	void sub(int i, int j, const fMatrix &m)
	{
		i *= 3;
		j *= 3;
		for (int k = 0; k < 3; ++k)
			for (int l = 0; l < 3; ++l)
				m_trips.push_back(Triplet(i + k, j + l, -m.coeff(l, k)));
	}

	inline void construct(lMatrix &m)
	{
		m.setFromTriplets(m_trips.begin(), m_trips.end());
		m_trips.clear();
	}

private:
	TripletList m_trips;
};

#ifdef USE_EIGEN_CORE
typedef Eigen::ConjugateGradient<lMatrix, Eigen::Lower, Eigen::DiagonalPreconditioner<Scalar> > ConjugateGradient;
#endif
#ifdef USE_EIGEN_CONSTRAINED_CG
typedef Eigen::ConstrainedConjugateGradient<lMatrix, Eigen::Lower, lMatrix,
                                            Eigen::DiagonalPreconditioner<Scalar> >
    ConstraintConjGrad;
#endif
using Eigen::ComputationInfo;

static void print_lvector(const lVector &v)
{
	for (int i = 0; i < v.rows(); ++i) {
		if (i > 0 && i % 3 == 0)
			printf("\n");

		printf("%f,\n", v[i]);
	}
}

static void print_lmatrix(const lMatrix &m)
{
	for (int j = 0; j < m.rows(); ++j) {
		if (j > 0 && j % 3 == 0)
			printf("\n");

		for (int i = 0; i < m.cols(); ++i) {
			if (i > 0 && i % 3 == 0)
				printf("  ");

			implicit_print_matrix_elem(m.coeff(j, i));
		}
		printf("\n");
	}
}

BLI_INLINE void lMatrix_reserve_elems(lMatrix &m, int num)
{
	m.reserve(Eigen::VectorXi::Constant(m.cols(), num));
}

BLI_INLINE float *lVector_v3(lVector &v, int vertex)
{
	return v.data() + 3 * vertex;
}

BLI_INLINE const float *lVector_v3(const lVector &v, int vertex)
{
	return v.data() + 3 * vertex;
}

#if 0
BLI_INLINE void triplets_m3(TripletList &tlist, float m[3][3], int i, int j)
{
	i *= 3;
	j *= 3;
	for (int l = 0; l < 3; ++l) {
		for (int k = 0; k < 3; ++k) {
			tlist.push_back(Triplet(i + k, j + l, m[k][l]));
		}
	}
}

BLI_INLINE void triplets_m3fl(TripletList &tlist, float m[3][3], int i, int j, float factor)
{
	i *= 3;
	j *= 3;
	for (int l = 0; l < 3; ++l) {
		for (int k = 0; k < 3; ++k) {
			tlist.push_back(Triplet(i + k, j + l, m[k][l] * factor));
		}
	}
}

BLI_INLINE void lMatrix_add_triplets(lMatrix &r, const TripletList &tlist)
{
	lMatrix t(r.rows(), r.cols());
	t.setFromTriplets(tlist.begin(), tlist.end());
	r += t;
}

BLI_INLINE void lMatrix_madd_triplets(lMatrix &r, const TripletList &tlist, float f)
{
	lMatrix t(r.rows(), r.cols());
	t.setFromTriplets(tlist.begin(), tlist.end());
	r += f * t;
}

BLI_INLINE void lMatrix_sub_triplets(lMatrix &r, const TripletList &tlist)
{
	lMatrix t(r.rows(), r.cols());
	t.setFromTriplets(tlist.begin(), tlist.end());
	r -= t;
}
#endif

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], 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], 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;
}

BLI_INLINE void madd_m3_m3m3fl(float r[3][3], float a[3][3], float b[3][3], float f)
{
	r[0][0] = a[0][0] + b[0][0] * f;
	r[0][1] = a[0][1] + b[0][1] * f;
	r[0][2] = a[0][2] + b[0][2] * f;
	r[1][0] = a[1][0] + b[1][0] * f;
	r[1][1] = a[1][1] + b[1][1] * f;
	r[1][2] = a[1][2] + b[1][2] * f;
	r[2][0] = a[2][0] + b[2][0] * f;
	r[2][1] = a[2][1] + b[2][1] * f;
	r[2][2] = a[2][2] + b[2][2] * f;
}

struct Implicit_Data {
	typedef std::vector<fMatrix> fMatrixVector;

	Implicit_Data(int numverts)
	{
		resize(numverts);
	}

	void resize(int numverts)
	{
		this->numverts = numverts;
		int tot = 3 * numverts;

		M.resize(tot, tot);
		F.resize(tot);
		dFdX.resize(tot, tot);
		dFdV.resize(tot, tot);

		tfm.resize(numverts, I);

		X.resize(tot);
		Xnew.resize(tot);
		V.resize(tot);
		Vnew.resize(tot);

		A.resize(tot, tot);
		B.resize(tot);

		dV.resize(tot);
		z.resize(tot);
		S.resize(tot, tot);

		iM.reserve(numverts);
		idFdX.reserve(numverts);
		idFdV.reserve(numverts);
		iS.reserve(numverts);
	}

	int numverts;

	/* inputs */
	lMatrix M;                  /* masses */
	lVector F;                  /* forces */
	lMatrix dFdX, dFdV;         /* force jacobians */

	fMatrixVector tfm;          /* local coordinate transform */

	/* motion state data */
	lVector X, Xnew;            /* positions */
	lVector V, Vnew;            /* velocities */

	/* internal solver data */
	lVector B;                  /* B for A*dV = B */
	lMatrix A;                  /* A for A*dV = B */

	lVector dV;                 /* velocity change (solution of A*dV = B) */
	lVector z;                  /* target velocity in constrained directions */
	lMatrix S;                  /* filtering matrix for constraints */

	/* temporary constructors */
	lMatrixCtor iM;             /* masses */
	lMatrixCtor idFdX, idFdV;   /* force jacobians */
	lMatrixCtor iS;             /* filtering matrix for constraints */
};

Implicit_Data *BPH_mass_spring_solver_create(int numverts, int numsprings)
{
	Implicit_Data *id = new Implicit_Data(numverts);
	return id;
}

void BPH_mass_spring_solver_free(Implicit_Data *id)
{
	if (id)
		delete id;
}

int BPH_mass_spring_solver_numvert(Implicit_Data *id)
{
	if (id)
		return id->numverts;
	else
		return 0;
}

/* ==== 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], 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], v);
}

BLI_INLINE void world_to_root_m3(Implicit_Data *data, int index, float r[3][3], float m[3][3])
{
	float trot[3][3];
	copy_m3_m3(trot, data->tfm[index]);
	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], float m[3][3])
{
	mul_m3_m3m3(r, data->tfm[index], m);
}

/* ================================ */

bool BPH_mass_spring_solve_velocities(Implicit_Data *data, float dt, ImplicitSolverResult *result)
{
#ifdef USE_EIGEN_CORE
	typedef ConjugateGradient solver_t;
#endif
#ifdef USE_EIGEN_CONSTRAINED_CG
	typedef ConstraintConjGrad solver_t;
#endif

	data->iM.construct(data->M);
	data->idFdX.construct(data->dFdX);
	data->idFdV.construct(data->dFdV);
	data->iS.construct(data->S);

	solver_t cg;
	cg.setMaxIterations(100);
	cg.setTolerance(0.01f);

#ifdef USE_EIGEN_CONSTRAINED_CG
	cg.filter() = data->S;
#endif

	data->A = data->M - dt * data->dFdV - dt * dt * data->dFdX;
	cg.compute(data->A);

	data->B = dt * data->F + dt * dt * data->dFdX * data->V;

#ifdef IMPLICIT_PRINT_SOLVER_INPUT_OUTPUT
	printf("==== A ====\n");
	print_lmatrix(id->A);
	printf("==== z ====\n");
	print_lvector(id->z);
	printf("==== B ====\n");
	print_lvector(id->B);
	printf("==== S ====\n");
	print_lmatrix(id->S);
#endif

#ifdef USE_EIGEN_CORE
	data->dV = cg.solve(data->B);
#endif
#ifdef USE_EIGEN_CONSTRAINED_CG
	data->dV = cg.solveWithGuess(data->B, data->z);
#endif

#ifdef IMPLICIT_PRINT_SOLVER_INPUT_OUTPUT
	printf("==== dV ====\n");
	print_lvector(id->dV);
	printf("========\n");
#endif

	data->Vnew = data->V + data->dV;

	switch (cg.info()) {
		case Eigen::Success:        result->status = BPH_SOLVER_SUCCESS;         break;
		case Eigen::NoConvergence:  result->status = BPH_SOLVER_NO_CONVERGENCE;  break;
		case Eigen::InvalidInput:   result->status = BPH_SOLVER_INVALID_INPUT;   break;
		case Eigen::NumericalIssue: result->status = BPH_SOLVER_NUMERICAL_ISSUE; break;
	}

	result->iterations = cg.iterations();
	result->error = cg.error();

	return cg.info() == Eigen::Success;
}

bool BPH_mass_spring_solve_positions(Implicit_Data *data, float dt)
{
	data->Xnew = data->X + data->Vnew * dt;
	return true;
}

/* ================================ */

void BPH_mass_spring_apply_result(Implicit_Data *data)
{
	data->X = data->Xnew;
	data->V = data->Vnew;
}

void BPH_mass_spring_set_vertex_mass(Implicit_Data *data, int index, float mass)
{
	float m[3][3];
	copy_m3_m3(m, I);
	mul_m3_fl(m, mass);
	data->iM.add(index, index, m);
}

void BPH_mass_spring_set_rest_transform(Implicit_Data *data, int index, float tfm[3][3])
{
#ifdef CLOTH_ROOT_FRAME
	copy_m3_m3(data->tfm[index], tfm);
#else
	unit_m3(data->tfm[index]);
	(void)tfm;
#endif
}

void BPH_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.v3(index), x);
	world_to_root_v3(data, index, data->V.v3(index), v);
}

void BPH_mass_spring_set_position(Implicit_Data *data, int index, const float x[3])
{
	world_to_root_v3(data, index, data->X.v3(index), x);
}

void BPH_mass_spring_set_velocity(Implicit_Data *data, int index, const float v[3])
{
	world_to_root_v3(data, index, data->V.v3(index), v);
}

void BPH_mass_spring_get_motion_state(struct Implicit_Data *data, int index, float x[3], float v[3])
{
	if (x) root_to_world_v3(data, index, x, data->X.v3(index));
	if (v) root_to_world_v3(data, index, v, data->V.v3(index));
}

void BPH_mass_spring_get_position(struct Implicit_Data *data, int index, float x[3])
{
	root_to_world_v3(data, index, x, data->X.v3(index));
}

void BPH_mass_spring_get_new_velocity(Implicit_Data *data, int index, float v[3])
{
	root_to_world_v3(data, index, v, data->V.v3(index));
}

void BPH_mass_spring_set_new_velocity(Implicit_Data *data, int index, const float v[3])
{
	world_to_root_v3(data, index, data->V.v3(index), v);
}

void BPH_mass_spring_clear_constraints(Implicit_Data *data)
{
	int numverts = data->numverts;
	for (int i = 0; i < numverts; ++i) {
		data->iS.add(i, i, I);
		zero_v3(data->z.v3(i));
	}
}

void BPH_mass_spring_add_constraint_ndof0(Implicit_Data *data, int index, const float dV[3])
{
	data->iS.sub(index, index, I);

	world_to_root_v3(data, index, data->z.v3(index), dV);
}

void BPH_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);
	outerproduct(cmat, p, p);
	copy_m3_m3(m, cmat);

	world_to_root_v3(data, index, q, c2);
	outerproduct(cmat, q, q);
	add_m3_m3m3(m, m, cmat);

	/* XXX not sure but multiplication should work here */
	data->iS.sub(index, index, 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.v3(index), u);
}

void BPH_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);
	outerproduct(cmat, p, p);
	copy_m3_m3(m, cmat);

	data->iS.sub(index, index, 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.v3(index), u);
}

void BPH_mass_spring_clear_forces(Implicit_Data *data)
{
	data->F.setZero();
	data->dFdX.setZero();
	data->dFdV.setZero();
}

void BPH_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.v3(index));
	cross_v3_v3v3(coriolis, w, data->V.v3(index));
	mul_v3_fl(coriolis, 2.0f);
	cross_v3_v3v3(rotvel, w, data->X.v3(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.v3(index), f);
	data->idFdX.add(index, index, dfdx);
	data->idFdV.add(index, index, dfdv);
#else
	(void)data;
	(void)index;
	(void)acceleration;
	(void)omega;
	(void)domega_dt;
#endif
}

void BPH_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.v3(index), f);
}

void BPH_mass_spring_force_drag(Implicit_Data *data, float drag)
{
	int numverts = data->numverts;
	for (int i = 0; i < numverts; i++) {
		float tmp[3][3];

		/* NB: uses root space velocity, no need to transform */
		madd_v3_v3fl(data->F.v3(i), data->V.v3(i), -drag);

		copy_m3_m3(tmp, I);
		mul_m3_fl(tmp, -drag);
		data->idFdV.add(i, i, tmp);
	}
}

void BPH_mass_spring_force_extern(struct 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.v3(i), tf);
	data->idFdX.add(i, i, tdfdx);
	data->idFdV.add(i, i, 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);
}

/* XXX does not support force jacobians yet, since the effector system does not provide them either */
void BPH_mass_spring_force_face_wind(Implicit_Data *data, int v1, int v2, int v3, const float(*winvec)[3])
{
	const float effector_scale = 0.02f;
	float win[3], nor[3], area;
	float factor;

	// calculate face normal and area
	area = calc_nor_area_tri(nor, data->X.v3(v1), data->X.v3(v2), data->X.v3(v3));
	factor = effector_scale * area / 3.0f;

	world_to_root_v3(data, v1, win, winvec[v1]);
	madd_v3_v3fl(data->F.v3(v1), nor, factor * dot_v3v3(win, nor));

	world_to_root_v3(data, v2, win, winvec[v2]);
	madd_v3_v3fl(data->F.v3(v2), nor, factor * dot_v3v3(win, nor));

	world_to_root_v3(data, v3, win, winvec[v3]);
	madd_v3_v3fl(data->F.v3(v3), nor, factor * dot_v3v3(win, nor));
}

void BPH_mass_spring_force_edge_wind(Implicit_Data *data, int v1, int v2, const float(*winvec)[3])
{
	const float effector_scale = 0.01;
	float win[3], dir[3], nor[3], length;

	sub_v3_v3v3(dir, data->X.v3(v1), data->X.v3(v2));
	length = normalize_v3(dir);

	world_to_root_v3(data, v1, win, winvec[v1]);
	madd_v3_v3v3fl(nor, win, dir, -dot_v3v3(win, dir));
	madd_v3_v3fl(data->F.v3(v1), nor, effector_scale * length);

	world_to_root_v3(data, v2, win, winvec[v2]);
	madd_v3_v3v3fl(nor, win, dir, -dot_v3v3(win, dir));
	madd_v3_v3fl(data->F.v3(v2), nor, effector_scale * length);
}

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) / MAX2(length, rest))));
}
#endif

BLI_INLINE void dfdv_damp(float to[3][3], const float dir[3], float damping)
{
	// derivative of force wrt velocity
	outerproduct(to, dir, dir);
	mul_m3_fl(to, -damping);
}

BLI_INLINE float fb(float length, float L)
{
	float x = length / L;
	return (-11.541f * powf(x, 4) + 34.193f * powf(x, 3) - 39.083f * powf(x, 2) + 23.116f * x - 9.713f);
}

BLI_INLINE float fbderiv(float length, float L)
{
	float x = length / L;

	return (-46.164f * powf(x, 3) + 102.579f * powf(x, 2) - 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;
	else
		return tempfb_fl;
}

// function to calculae 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;
	}
	else {
		return -kb *fbderiv(length, L);
	}
}

/* calculate elonglation */
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.v3(j), data->X.v3(i));
	sub_v3_v3v3(r_vel, data->V.v3(j), data->V.v3(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], float dfdx[3][3], float dfdv[3][3])
{
	add_v3_v3(data->F.v3(i), f);
	sub_v3_v3(data->F.v3(j), f);

	data->idFdX.add(i, i, dfdx);
	data->idFdX.add(j, j, dfdx);
	data->idFdX.sub(i, j, dfdx);
	data->idFdX.sub(j, i, dfdx);

	data->idFdV.add(i, i, dfdv);
	data->idFdV.add(j, j, dfdv);
	data->idFdV.sub(i, j, dfdv);
	data->idFdV.sub(j, i, dfdv);
}

bool BPH_mass_spring_force_spring_linear(Implicit_Data *data, int i, int j, float restlen,
                                         float stiffness, float damping, bool no_compress, float clamp_force,
                                         float r_f[3], float r_dfdx[3][3], float r_dfdv[3][3])
{
	float extent[3], length, dir[3], vel[3];

	// calculate elonglation
	spring_length(data, i, j, extent, dir, &length, vel);

	if (length > restlen || no_compress) {
		float stretch_force, f[3], dfdx[3][3], dfdv[3][3];

		stretch_force = stiffness * (length - restlen);
		if (clamp_force > 0.0f && stretch_force > clamp_force) {
			stretch_force = clamp_force;
		}
		mul_v3_v3fl(f, dir, stretch_force);

		// Ascher & Boxman, p.21: Damping only during elonglation
		// something wrong with it...
		madd_v3_v3fl(f, dir, damping * dot_v3v3(vel, dir));

		dfdx_spring(dfdx, dir, length, restlen, stiffness);
		dfdv_damp(dfdv, dir, damping);

		apply_spring(data, i, j, f, dfdx, dfdv);

		if (r_f) copy_v3_v3(r_f, f);
		if (r_dfdx) copy_m3_m3(r_dfdx, dfdx);
		if (r_dfdv) copy_m3_m3(r_dfdv, dfdv);

		return true;
	}
	else {
		if (r_f) zero_v3(r_f);
		if (r_dfdx) zero_m3(r_dfdx);
		if (r_dfdv) zero_m3(r_dfdv);

		return false;
	}
}

/* See "Stable but Responsive Cloth" (Choi, Ko 2005) */
bool BPH_mass_spring_force_spring_bending(Implicit_Data *data, int i, int j, float restlen,
                                          float kb, float cb,
                                          float r_f[3], float r_dfdx[3][3], float r_dfdv[3][3])
{
	float extent[3], length, dir[3], vel[3];

	// calculate elonglation
	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);

		if (r_f) copy_v3_v3(r_f, f);
		if (r_dfdx) copy_m3_m3(r_dfdx, dfdx);
		if (r_dfdv) copy_m3_m3(r_dfdv, dfdv);

		return true;
	}
	else {
		if (r_f) zero_v3(r_f);
		if (r_dfdx) zero_m3(r_dfdx);
		if (r_dfdv) zero_m3(r_dfdv);

		return false;
	}
}

/* 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.v3(j), data->X.v3(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_angbend_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.v3(j), data->X.v3(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.v3(k), data->X.v3(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.v3(j), data->V.v3(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.v3(k), data->V.v3(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_angbend_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_angbend_forces(data, i, j, k, goal, stiffness, damping,
		                      q, dvec_pos[a], dvec_null[a], f);
		copy_v3_v3(dfdx[a], f);

		spring_angbend_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_angbend_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_angbend_forces(data, i, j, k, goal, stiffness, damping,
		                      q, dvec_null[a], dvec_pos[a], f);
		copy_v3_v3(dfdv[a], f);

		spring_angbend_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;
		}
	}
}

/* Angular spring that pulls the vertex toward the local target
 * See "Artistic Simulation of Curly Hair" (Pixar technical memo #12-03a)
 */
bool BPH_mass_spring_force_spring_bending_angular(Implicit_Data *data, int i, int j, int k,
                                                  const float target[3], float stiffness, float damping)
{
	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};

	world_to_root_v3(data, j, goal, target);

	spring_angbend_forces(data, i, j, k, goal, stiffness, damping, k, vecnull, vecnull, fk);
	negate_v3_v3(fj, fk); /* counterforce */

	spring_angbend_estimate_dfdx(data, i, j, k, goal, stiffness, damping, i, dfk_dxi);
	spring_angbend_estimate_dfdx(data, i, j, k, goal, stiffness, damping, j, dfk_dxj);
	spring_angbend_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_angbend_estimate_dfdv(data, i, j, k, goal, stiffness, damping, i, dfk_dvi);
	spring_angbend_estimate_dfdv(data, i, j, k, goal, stiffness, damping, j, dfk_dvj);
	spring_angbend_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.v3(j), fj);
	add_v3_v3(data->F.v3(k), fk);

	data->idFdX.add(j, j, dfj_dxj);
	data->idFdX.add(k, k, dfk_dxk);

	data->idFdX.add(i, j, dfj_dxi);
	data->idFdX.add(j, i, dfj_dxi);
	data->idFdX.add(j, k, dfk_dxj);
	data->idFdX.add(k, j, dfk_dxj);
	data->idFdX.add(i, k, dfk_dxi);
	data->idFdX.add(k, i, dfk_dxi);

	data->idFdV.add(j, j, dfj_dvj);
	data->idFdV.add(k, k, dfk_dvk);

	data->idFdV.add(i, j, dfj_dvi);
	data->idFdV.add(j, i, dfj_dvi);
	data->idFdV.add(j, k, dfk_dvj);
	data->idFdV.add(k, j, dfk_dvj);
	data->idFdV.add(i, k, dfk_dvi);
	data->idFdV.add(k, i, 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 BPH_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 r_f[3], float r_dfdx[3][3], float r_dfdv[3][3])
{
	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.v3(i));
	sub_v3_v3v3(vel, root_goal_v, data->V.v3(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 elonglation
		// 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.v3(i), f);
		data->idFdX.add(i, i, dfdx);
		data->idFdV.add(i, i, dfdv);

		if (r_f) copy_v3_v3(r_f, f);
		if (r_dfdx) copy_m3_m3(r_dfdx, dfdx);
		if (r_dfdv) copy_m3_m3(r_dfdv, dfdv);

		return true;
	}
	else {
		if (r_f) zero_v3(r_f);
		if (r_dfdx) zero_m3(r_dfdx);
		if (r_dfdv) zero_m3(r_dfdv);

		return false;
	}
}

#endif /* IMPLICIT_SOLVER_EIGEN */