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test_elastic_materials.cc
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			/* -------------------------------------------------------------------------- */
			#include "material_elastic.hh"
			#include "material_elastic_orthotropic.hh"
			#include "solid_mechanics_model.hh"
			#include "test_material_fixtures.hh"
			/* -------------------------------------------------------------------------- */
			#include <gtest/gtest.h>
			#include <type_traits>
			/* -------------------------------------------------------------------------- */

			using namespace akantu;

			using types =
			::testing::Types<Traits<MaterialElastic, 1>, Traits<MaterialElastic, 2>,
			Traits<MaterialElastic, 3>,

			Traits<MaterialElasticOrthotropic, 2>,
			Traits<MaterialElasticOrthotropic, 3>,

			Traits<MaterialElasticLinearAnisotropic, 2>,
			Traits<MaterialElasticLinearAnisotropic, 3>>;

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElastic<1>>::testComputeStress() {
			Real E = 3.;
			setParam("E", E);

			Matrix<Real> eps = {{2}};
			Matrix<Real> sigma(1, 1);
			Real sigma_th = 2;
			this->computeStressOnQuad(eps, sigma, sigma_th);

			auto solution = E * eps(0, 0) + sigma_th;
			ASSERT_NEAR(sigma(0, 0), solution, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElastic<1>>::testEnergyDensity() {
			Real eps = 2, sigma = 2;
			Real epot = 0;
			this->computePotentialEnergyOnQuad({{eps}}, {{sigma}}, epot);
			Real solution = 2;
			ASSERT_NEAR(epot, solution, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <>
			void FriendMaterial<MaterialElastic<1>>::testComputeTangentModuli() {
			Real E = 2;
			setParam("E", E);
			Matrix<Real> tangent(1, 1);
			this->computeTangentModuliOnQuad(tangent);
			ASSERT_NEAR(tangent(0, 0), E, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElastic<1>>::testPushWaveSpeed() {
			Real E = 3., rho = 2.;
			setParam("E", E);
			setParam("rho", rho);

			auto wave_speed = this->getPushWaveSpeed(Element());
			auto solution = std::sqrt(E / rho);
			ASSERT_NEAR(wave_speed, solution, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElastic<1>>::testShearWaveSpeed() {
			ASSERT_THROW(this->getShearWaveSpeed(Element()), debug::Exception);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElastic<2>>::testComputeStress() {
			Real E = 1.;
			Real nu = .3;
			Real sigma_th = 0.3; // thermal stress
			setParam("E", E);
			setParam("nu", nu);

			Matrix<Real> rotation_matrix = getRandomRotation2d();

			auto grad_u = this->getDeviatoricStrain(1.).block(0, 0, 2, 2);

			auto grad_u_rot = this->applyRotation(grad_u, rotation_matrix);

			Matrix<Real> sigma_rot(2, 2);
			this->computeStressOnQuad(grad_u_rot, sigma_rot, sigma_th);

			auto sigma = this->reverseRotation(sigma_rot, rotation_matrix);

			Matrix<Real> identity(2, 2);
			identity.eye();

			Matrix<Real> sigma_expected =
			0.5 * E / (1 + nu) * (grad_u + grad_u.transpose()) + sigma_th * identity;

			auto diff = sigma - sigma_expected;
			Real stress_error = diff.norm<L_inf>();
			ASSERT_NEAR(stress_error, 0., 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElastic<2>>::testEnergyDensity() {
			Matrix<Real> sigma = {{1, 2}, {2, 4}};
			Matrix<Real> eps = {{1, 0}, {0, 1}};
			Real epot = 0;
			Real solution = 2.5;
			this->computePotentialEnergyOnQuad(eps, sigma, epot);
			ASSERT_NEAR(epot, solution, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <>
			void FriendMaterial<MaterialElastic<2>>::testComputeTangentModuli() {
			Real E = 1.;
			Real nu = .3;
			setParam("E", E);
			setParam("nu", nu);
			Matrix<Real> tangent(3, 3);

			/* Plane Strain */
			// clang-format off
			Matrix<Real> solution = {
			{1 - nu, nu, 0},
			{nu, 1 - nu, 0},
			{0, 0, (1 - 2 * nu) / 2},
			};
			// clang-format on
			solution = E / ((1 + nu) (1 - 2 * nu));

			this->computeTangentModuliOnQuad(tangent);
			Real tangent_error = (tangent - solution).norm<L_2>();
			ASSERT_NEAR(tangent_error, 0, 1e-14);

			/* Plane Stress */
			this->plane_stress = true;
			this->updateInternalParameters();
			// clang-format off
			solution = {
			{1, nu, 0},
			{nu, 1, 0},
			{0, 0, (1 - nu) / 2},
			};
			// clang-format on
			solution = E / (1 - nu nu);

			this->computeTangentModuliOnQuad(tangent);
			tangent_error = (tangent - solution).norm<L_2>();
			ASSERT_NEAR(tangent_error, 0, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElastic<2>>::testPushWaveSpeed() {
			Real E = 1.;
			Real nu = .3;
			Real rho = 2;
			setParam("E", E);
			setParam("nu", nu);
			setParam("rho", rho);
			auto wave_speed = this->getPushWaveSpeed(Element());

			Real K = E / (3 * (1 - 2 * nu));
			Real mu = E / (2 * (1 + nu));
			Real sol = std::sqrt((K + 4. / 3 * mu) / rho);

			ASSERT_NEAR(wave_speed, sol, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElastic<2>>::testShearWaveSpeed() {
			Real E = 1.;
			Real nu = .3;
			Real rho = 2;
			setParam("E", E);
			setParam("nu", nu);
			setParam("rho", rho);
			auto wave_speed = this->getShearWaveSpeed(Element());

			Real mu = E / (2 * (1 + nu));
			Real sol = std::sqrt(mu / rho);

			ASSERT_NEAR(wave_speed, sol, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElastic<3>>::testComputeStress() {
			Real E = 1.;
			Real nu = .3;
			Real sigma_th = 0.3; // thermal stress
			setParam("E", E);
			setParam("nu", nu);

			Matrix<Real> rotation_matrix = getRandomRotation3d();

			auto grad_u = this->getDeviatoricStrain(1.);

			auto grad_u_rot = this->applyRotation(grad_u, rotation_matrix);

			Matrix<Real> sigma_rot(3, 3);
			this->computeStressOnQuad(grad_u_rot, sigma_rot, sigma_th);

			auto sigma = this->reverseRotation(sigma_rot, rotation_matrix);

			Matrix<Real> identity(3, 3);
			identity.eye();

			Matrix<Real> sigma_expected =
			0.5 * E / (1 + nu) * (grad_u + grad_u.transpose()) + sigma_th * identity;

			auto diff = sigma - sigma_expected;
			Real stress_error = diff.norm<L_inf>();
			ASSERT_NEAR(stress_error, 0., 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElastic<3>>::testEnergyDensity() {
			Matrix<Real> sigma = {{1, 2, 3}, {2, 4, 5}, {3, 5, 6}};
			Matrix<Real> eps = {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}};
			Real epot = 0;
			Real solution = 5.5;
			this->computePotentialEnergyOnQuad(eps, sigma, epot);
			ASSERT_NEAR(epot, solution, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <>
			void FriendMaterial<MaterialElastic<3>>::testComputeTangentModuli() {
			Real E = 1.;
			Real nu = .3;
			setParam("E", E);
			setParam("nu", nu);
			Matrix<Real> tangent(6, 6);

			// clang-format off
			Matrix<Real> solution = {
			{1 - nu, nu, nu, 0, 0, 0},
			{nu, 1 - nu, nu, 0, 0, 0},
			{nu, nu, 1 - nu, 0, 0, 0},
			{0, 0, 0, (1 - 2 * nu) / 2, 0, 0},
			{0, 0, 0, 0, (1 - 2 * nu) / 2, 0},
			{0, 0, 0, 0, 0, (1 - 2 * nu) / 2},
			};
			// clang-format on
			solution = E / ((1 + nu) (1 - 2 * nu));

			this->computeTangentModuliOnQuad(tangent);
			Real tangent_error = (tangent - solution).norm<L_2>();
			ASSERT_NEAR(tangent_error, 0, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElastic<3>>::testPushWaveSpeed() {
			Real E = 1.;
			Real nu = .3;
			Real rho = 2;
			setParam("E", E);
			setParam("nu", nu);
			setParam("rho", rho);
			auto wave_speed = this->getPushWaveSpeed(Element());

			Real K = E / (3 * (1 - 2 * nu));
			Real mu = E / (2 * (1 + nu));
			Real sol = std::sqrt((K + 4. / 3 * mu) / rho);

			ASSERT_NEAR(wave_speed, sol, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElastic<3>>::testShearWaveSpeed() {
			Real E = 1.;
			Real nu = .3;
			Real rho = 2;
			setParam("E", E);
			setParam("nu", nu);
			setParam("rho", rho);
			auto wave_speed = this->getShearWaveSpeed(Element());

			Real mu = E / (2 * (1 + nu));
			Real sol = std::sqrt(mu / rho);

			ASSERT_NEAR(wave_speed, sol, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElasticOrthotropic<2>>::testComputeStress() {
			UInt Dim = 2;
			Real E1 = 1.;
			Real E2 = 2.;
			Real nu12 = 0.1;
			Real G12 = 2.;

			setParamNoUpdate("E1", E1);
			setParamNoUpdate("E2", E2);
			setParamNoUpdate("nu12", nu12);
			setParamNoUpdate("G12", G12);

			// material frame of reference is rotate by rotation_matrix starting from
			// canonical basis
			Matrix<Real> rotation_matrix = getRandomRotation2d();

			// canonical basis as expressed in the material frame of reference, as
			// required by MaterialElasticOrthotropic class (it is simply given by the
			// columns of the rotation_matrix; the lines give the material basis expressed
			// in the canonical frame of reference)
			Vector<Real> v1(Dim);
			Vector<Real> v2(Dim);
			for (UInt i = 0; i < Dim; ++i) {
			v1[i] = rotation_matrix(i, 0);
			v2[i] = rotation_matrix(i, 1);
			}

			setParamNoUpdate("n1", v1.normalize());
			setParamNoUpdate("n2", v2.normalize());

			// set internal Cijkl matrix expressed in the canonical frame of reference
			this->updateInternalParameters();

			// gradient in material frame of reference
			auto grad_u = ( this->getDeviatoricStrain(2.) + this->getHydrostaticStrain(1.) ).block(0, 0, 2, 2);

			// gradient in canonical basis (we need to rotate back to the canonical
			// basis)
			auto grad_u_rot = this->reverseRotation(grad_u, rotation_matrix);

			// stress in the canonical basis
			Matrix<Real> sigma_rot(2, 2);
			this->computeStressOnQuad(grad_u_rot, sigma_rot);

			// stress in the material reference (we need to apply the rotation)
			auto sigma = this->applyRotation(sigma_rot, rotation_matrix);

			// construction of Cijkl engineering tensor in the material frame of reference
			// ref: http://solidmechanics.org/Text/Chapter3_2/Chapter3_2.php#Sect3_2_13
			Real nu21 = nu12*E2/E1;
			Real gamma = 1 / ( 1 - nu12*nu21 );

			Matrix<Real> C_expected(2Dim,2Dim,0);
			C_expected(0,0) = gamma * E1 ;
			C_expected(1,1) = gamma * E2 ;
			C_expected(2,2) = G12;

			C_expected(1,0) = C_expected(0,1) = gamma * E1 * nu21;

			// epsilon is computed directly in the material frame of reference
			Matrix<Real> epsilon = 0.5 * ( grad_u + grad_u.transpose() );

			// sigma_expected is computed directly in the material frame of reference
			Matrix<Real> sigma_expected(Dim, Dim);
			for ( UInt i = 0; i < Dim; ++i ) {
			for ( UInt j = 0; j < Dim; ++j ) {
			sigma_expected(i,i) += C_expected(i,j)*epsilon(j,j);
			}
			}

			sigma_expected(0,1) = sigma_expected(1,0) = C_expected(2,2) * 2 * epsilon(0,1);

			// sigmas are checked in the material frame of reference
			auto diff = sigma - sigma_expected;
			Real stress_error = diff.norm<L_inf>();
			ASSERT_NEAR(stress_error, 0., 1e-13);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElasticOrthotropic<2>>::testEnergyDensity() {
			Matrix<Real> sigma = {{1, 2}, {2, 4}};
			Matrix<Real> eps = {{1, 0}, {0, 1}};
			Real epot = 0;
			Real solution = 2.5;
			this->computePotentialEnergyOnQuad(eps, sigma, epot);
			ASSERT_NEAR(epot, solution, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElasticOrthotropic<2>>::testComputeTangentModuli() {

			// Note: for this test material and canonical basis coincide
			Vector<Real> n1 = {1, 0};
			Vector<Real> n2 = {0, 1};
			Real E1 = 1.;
			Real E2 = 2.;
			Real nu12 = 0.1;
			Real G12 = 2.;

			setParamNoUpdate("n1", n1);
			setParamNoUpdate("n2", n2);
			setParamNoUpdate("E1", E1);
			setParamNoUpdate("E2", E2);
			setParamNoUpdate("nu12", nu12);
			setParamNoUpdate("G12", G12);

			// set internal Cijkl matrix expressed in the canonical frame of reference
			this->updateInternalParameters();

			// construction of Cijkl engineering tensor in the material frame of reference
			// ref: http://solidmechanics.org/Text/Chapter3_2/Chapter3_2.php#Sect3_2_13
			Real nu21 = nu12*E2/E1;
			Real gamma = 1 / ( 1 - nu12*nu21 );

			Matrix<Real> C_expected(3,3);
			C_expected(0,0) = gamma * E1 ;
			C_expected(1,1) = gamma * E2 ;
			C_expected(2,2) = G12;

			C_expected(1,0) = C_expected(0,1) = gamma * E1 * nu21;

			Matrix<Real> tangent(3,3);
			this->computeTangentModuliOnQuad(tangent);

			Real tangent_error = (tangent - C_expected).norm<L_2>();
			ASSERT_NEAR(tangent_error, 0, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElasticOrthotropic<2>>::testPushWaveSpeed() {

			// Note: for this test material and canonical basis coincide
			Vector<Real> n1 = {1, 0};
			Vector<Real> n2 = {0, 1};
			Real E1 = 1.;
			Real E2 = 2.;
			Real nu12 = 0.1;
			Real G12 = 2.;
			Real rho = 2.5;

			setParamNoUpdate("n1", n1);
			setParamNoUpdate("n2", n2);
			setParamNoUpdate("E1", E1);
			setParamNoUpdate("E2", E2);
			setParamNoUpdate("nu12", nu12);
			setParamNoUpdate("G12", G12);
			setParamNoUpdate("rho", rho);

			// set internal Cijkl matrix expressed in the canonical frame of reference
			this->updateInternalParameters();

			// construction of Cijkl engineering tensor in the material frame of reference
			// ref: http://solidmechanics.org/Text/Chapter3_2/Chapter3_2.php#Sect3_2_13
			Real nu21 = nu12*E2/E1;
			Real gamma = 1 / ( 1 - nu12*nu21 );

			Matrix<Real> C_expected(3,3);
			C_expected(0,0) = gamma * E1 ;
			C_expected(1,1) = gamma * E2 ;
			C_expected(2,2) = G12;

			C_expected(1,0) = C_expected(0,1) = gamma * E1 * nu21;

			Vector<Real> eig_expected(3);
			C_expected.eig(eig_expected);

			auto celerity_expected = std::sqrt(eig_expected(0)/rho);

			auto celerity = this->getCelerity(Element());

			ASSERT_NEAR(celerity_expected, celerity, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElasticOrthotropic<2>>::testShearWaveSpeed() {

			AKANTU_DEBUG_WARNING("WARNING: Shear wave speed for 2D orthotropic materials is not present in akantu."
			"Celerity is used instead to get the stable time step (see testPushWaveSpeed).");
			ASSERT_NEAR(0, 0, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElasticOrthotropic<3>>::testComputeStress() {
			UInt Dim = 3;
			Real E1 = 1.;
			Real E2 = 2.;
			Real E3 = 3.;
			Real nu12 = 0.1;
			Real nu13 = 0.2;
			Real nu23 = 0.3;
			Real G12 = 2.;
			Real G13 = 3.;
			Real G23 = 1.;

			setParamNoUpdate("E1", E1);
			setParamNoUpdate("E2", E2);
			setParamNoUpdate("E3", E3);
			setParamNoUpdate("nu12", nu12);
			setParamNoUpdate("nu13", nu13);
			setParamNoUpdate("nu23", nu23);
			setParamNoUpdate("G12", G12);
			setParamNoUpdate("G13", G13);
			setParamNoUpdate("G23", G23);

			// material frame of reference is rotate by rotation_matrix starting from
			// canonical basis
			Matrix<Real> rotation_matrix = getRandomRotation3d();

			// canonical basis as expressed in the material frame of reference, as
			// required by MaterialElasticOrthotropic class (it is simply given by the
			// columns of the rotation_matrix; the lines give the material basis expressed
			// in the canonical frame of reference)
			Vector<Real> v1(Dim);
			Vector<Real> v2(Dim);
			Vector<Real> v3(Dim);
			for (UInt i = 0; i < Dim; ++i) {
			v1[i] = rotation_matrix(i, 0);
			v2[i] = rotation_matrix(i, 1);
			v3[i] = rotation_matrix(i, 2);
			}

			setParamNoUpdate("n1", v1.normalize());
			setParamNoUpdate("n2", v2.normalize());
			setParamNoUpdate("n3", v3.normalize());

			// set internal Cijkl matrix expressed in the canonical frame of reference
			this->updateInternalParameters();

			// gradient in material frame of reference
			auto grad_u = this->getDeviatoricStrain(2.) + this->getHydrostaticStrain(1.);

			// gradient in canonical basis (we need to rotate back to the canonical
			// basis)
			auto grad_u_rot = this->reverseRotation(grad_u, rotation_matrix);

			// stress in the canonical basis
			Matrix<Real> sigma_rot(3, 3);
			this->computeStressOnQuad(grad_u_rot, sigma_rot);

			// stress in the material reference (we need to apply the rotation)
			auto sigma = this->applyRotation(sigma_rot, rotation_matrix);

			// construction of Cijkl engineering tensor in the material frame of reference
			// ref: http://solidmechanics.org/Text/Chapter3_2/Chapter3_2.php#Sect3_2_13
			Real nu21 = nu12*E2/E1;
			Real nu31 = nu13*E3/E1;
			Real nu32 = nu23*E3/E2;
			Real gamma = 1 / ( 1 - nu12nu21 - nu23nu32 - nu31nu13 - 2nu21nu32nu13 );

			Matrix<Real> C_expected(6,6);
			C_expected(0,0) = gamma * E1 * ( 1-nu23*nu32 );
			C_expected(1,1) = gamma * E2 * ( 1-nu13*nu31 );
			C_expected(2,2) = gamma * E3 * ( 1-nu12*nu21 );

			C_expected(1,0) = C_expected(0,1) = gamma * E1 * ( nu21 + nu31*nu23 );
			C_expected(2,0) = C_expected(0,2) = gamma * E1 * ( nu31 + nu21*nu32 );
			C_expected(2,1) = C_expected(1,2) = gamma * E2 * ( nu32 + nu12*nu31 );

			C_expected(3,3) = G23;
			C_expected(4,4) = G13;
			C_expected(5,5) = G12;

			// epsilon is computed directly in the material frame of reference
			Matrix<Real> epsilon = 0.5 * ( grad_u + grad_u.transpose() );

			// sigma_expected is computed directly in the material frame of reference
			Matrix<Real> sigma_expected(Dim, Dim);
			for ( UInt i = 0; i < Dim; ++i ) {
			for ( UInt j = 0; j < Dim; ++j ) {
			sigma_expected(i,i) += C_expected(i,j)*epsilon(j,j);
			}
			}

			sigma_expected(0,1) = C_expected(5,5) * 2 * epsilon(0,1);
			sigma_expected(0,2) = C_expected(4,4) * 2 * epsilon(0,2);
			sigma_expected(1,2) = C_expected(3,3) * 2 * epsilon(1,2);
			sigma_expected(1,0) = sigma_expected(0,1);
			sigma_expected(2,0) = sigma_expected(0,2);
			sigma_expected(2,1) = sigma_expected(1,2);

			// sigmas are checked in the material frame of reference
			auto diff = sigma - sigma_expected;
			Real stress_error = diff.norm<L_inf>();
			ASSERT_NEAR(stress_error, 0., 1e-13);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElasticOrthotropic<3>>::testEnergyDensity() {
			Matrix<Real> sigma = {{1, 2, 3}, {2, 4, 5}, {3, 5, 6}};
			Matrix<Real> eps = {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}};
			Real epot = 0;
			Real solution = 5.5;
			this->computePotentialEnergyOnQuad(eps, sigma, epot);
			ASSERT_NEAR(epot, solution, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElasticOrthotropic<3>>::testComputeTangentModuli() {

			// Note: for this test material and canonical basis coincide
			UInt Dim = 3;
			Vector<Real> n1 = {1, 0, 0};
			Vector<Real> n2 = {0, 1, 0};
			Vector<Real> n3 = {0, 0, 1};
			Real E1 = 1.;
			Real E2 = 2.;
			Real E3 = 3.;
			Real nu12 = 0.1;
			Real nu13 = 0.2;
			Real nu23 = 0.3;
			Real G12 = 2.;
			Real G13 = 3.;
			Real G23 = 1.;

			setParamNoUpdate("n1", n1);
			setParamNoUpdate("n2", n2);
			setParamNoUpdate("n3", n3);
			setParamNoUpdate("E1", E1);
			setParamNoUpdate("E2", E2);
			setParamNoUpdate("E3", E3);
			setParamNoUpdate("nu12", nu12);
			setParamNoUpdate("nu13", nu13);
			setParamNoUpdate("nu23", nu23);
			setParamNoUpdate("G12", G12);
			setParamNoUpdate("G13", G13);
			setParamNoUpdate("G23", G23);

			// set internal Cijkl matrix expressed in the canonical frame of reference
			this->updateInternalParameters();

			// construction of Cijkl engineering tensor in the material frame of reference
			// ref: http://solidmechanics.org/Text/Chapter3_2/Chapter3_2.php#Sect3_2_13
			Real nu21 = nu12*E2/E1;
			Real nu31 = nu13*E3/E1;
			Real nu32 = nu23*E3/E2;
			Real gamma = 1 / ( 1 - nu12nu21 - nu23nu32 - nu31nu13 - 2nu21nu32nu13 );

			Matrix<Real> C_expected(2Dim,2Dim,0);
			C_expected(0,0) = gamma * E1 * ( 1-nu23*nu32 );
			C_expected(1,1) = gamma * E2 * ( 1-nu13*nu31 );
			C_expected(2,2) = gamma * E3 * ( 1-nu12*nu21 );

			C_expected(1,0) = C_expected(0,1) = gamma * E1 * ( nu21 + nu31*nu23 );
			C_expected(2,0) = C_expected(0,2) = gamma * E1 * ( nu31 + nu21*nu32 );
			C_expected(2,1) = C_expected(1,2) = gamma * E2 * ( nu32 + nu12*nu31 );

			C_expected(3,3) = G23;
			C_expected(4,4) = G13;
			C_expected(5,5) = G12;

			Matrix<Real> tangent(6,6);
			this->computeTangentModuliOnQuad(tangent);

			Real tangent_error = (tangent - C_expected).norm<L_2>();
			ASSERT_NEAR(tangent_error, 0, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElasticOrthotropic<3>>::testPushWaveSpeed() {

			// Note: for this test material and canonical basis coincide
			UInt Dim = 3;
			Vector<Real> n1 = {1, 0, 0};
			Vector<Real> n2 = {0, 1, 0};
			Vector<Real> n3 = {0, 0, 1};
			Real E1 = 1.;
			Real E2 = 2.;
			Real E3 = 3.;
			Real nu12 = 0.1;
			Real nu13 = 0.2;
			Real nu23 = 0.3;
			Real G12 = 2.;
			Real G13 = 3.;
			Real G23 = 1.;
			Real rho = 2.3;

			setParamNoUpdate("n1", n1);
			setParamNoUpdate("n2", n2);
			setParamNoUpdate("n3", n3);
			setParamNoUpdate("E1", E1);
			setParamNoUpdate("E2", E2);
			setParamNoUpdate("E3", E3);
			setParamNoUpdate("nu12", nu12);
			setParamNoUpdate("nu13", nu13);
			setParamNoUpdate("nu23", nu23);
			setParamNoUpdate("G12", G12);
			setParamNoUpdate("G13", G13);
			setParamNoUpdate("G23", G23);
			setParamNoUpdate("rho", rho);

			// set internal Cijkl matrix expressed in the canonical frame of reference
			this->updateInternalParameters();

			// construction of Cijkl engineering tensor in the material frame of reference
			// ref: http://solidmechanics.org/Text/Chapter3_2/Chapter3_2.php#Sect3_2_13
			Real nu21 = nu12*E2/E1;
			Real nu31 = nu13*E3/E1;
			Real nu32 = nu23*E3/E2;
			Real gamma = 1 / ( 1 - nu12nu21 - nu23nu32 - nu31nu13 - 2nu21nu32nu13 );

			Matrix<Real> C_expected(2Dim,2Dim,0);
			C_expected(0,0) = gamma * E1 * ( 1-nu23*nu32 );
			C_expected(1,1) = gamma * E2 * ( 1-nu13*nu31 );
			C_expected(2,2) = gamma * E3 * ( 1-nu12*nu21 );

			C_expected(1,0) = C_expected(0,1) = gamma * E1 * ( nu21 + nu31*nu23 );
			C_expected(2,0) = C_expected(0,2) = gamma * E1 * ( nu31 + nu21*nu32 );
			C_expected(2,1) = C_expected(1,2) = gamma * E2 * ( nu32 + nu12*nu31 );

			C_expected(3,3) = G23;
			C_expected(4,4) = G13;
			C_expected(5,5) = G12;

			Vector<Real> eig_expected(6);
			C_expected.eig(eig_expected);

			auto celerity_expected = std::sqrt(eig_expected(0)/rho);

			auto celerity = this->getCelerity(Element());

			ASSERT_NEAR(celerity_expected, celerity, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElasticOrthotropic<3>>::testShearWaveSpeed() {

			AKANTU_DEBUG_WARNING("WARNING: Shear wave speed for 3D orthotropic materials is not present in akantu."
			"Celerity is used instead to get the stable time step (see testPushWaveSpeed).");
			ASSERT_NEAR(0, 0, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElasticLinearAnisotropic<2>>::testComputeStress() {
			UInt Dim = 2;
			Matrix<Real> C = {
			{1.0, 0.3, 0.4},
			{0.3, 2.0, 0.1},
			{0.4, 0.1, 1.5},
			};

			setParamNoUpdate("C11", C(0,0));
			setParamNoUpdate("C12", C(0,1));
			setParamNoUpdate("C13", C(0,2));
			setParamNoUpdate("C22", C(1,1));
			setParamNoUpdate("C23", C(1,2));
			setParamNoUpdate("C33", C(2,2));

			// material frame of reference is rotate by rotation_matrix starting from
			// canonical basis
			Matrix<Real> rotation_matrix = getRandomRotation2d();

			// canonical basis as expressed in the material frame of reference, as
			// required by MaterialElasticLinearAnisotropic class (it is simply given by
			// the columns of the rotation_matrix; the lines give the material basis
			// expressed in the canonical frame of reference)
			Vector<Real> v1(Dim);
			Vector<Real> v2(Dim);
			for (UInt i = 0; i < Dim; ++i) {
			v1[i] = rotation_matrix(i, 0);
			v2[i] = rotation_matrix(i, 1);
			}

			setParamNoUpdate("n1", v1.normalize());
			setParamNoUpdate("n2", v2.normalize());

			// set internal Cijkl matrix expressed in the canonical frame of reference
			this->updateInternalParameters();

			// gradient in material frame of reference
			auto grad_u = (this->getDeviatoricStrain(2.) + this->getHydrostaticStrain(1.)).block(0, 0, 2, 2);

			// gradient in canonical basis (we need to rotate back to the canonical
			// basis)
			auto grad_u_rot = this->reverseRotation(grad_u, rotation_matrix);

			// stress in the canonical basis
			Matrix<Real> sigma_rot(2, 2);
			this->computeStressOnQuad(grad_u_rot, sigma_rot);

			// stress in the material reference (we need to apply the rotation)
			auto sigma = this->applyRotation(sigma_rot, rotation_matrix);

			// epsilon is computed directly in the material frame of reference
			Matrix<Real> epsilon = 0.5 * ( grad_u + grad_u.transpose() );
			Vector<Real> epsilon_voigt(3);
			epsilon_voigt(0) = epsilon(0,0);
			epsilon_voigt(1) = epsilon(1,1);
			epsilon_voigt(2) = 2 * epsilon(0,1);

			// sigma_expected is computed directly in the material frame of reference
			Vector<Real> sigma_voigt = C * epsilon_voigt;
			Matrix<Real> sigma_expected(Dim, Dim);
			sigma_expected(0,0) = sigma_voigt(0);
			sigma_expected(1,1) = sigma_voigt(1);
			sigma_expected(0,1) = sigma_expected(1,0) = sigma_voigt(2);

			// sigmas are checked in the material frame of reference
			auto diff = sigma - sigma_expected;
			Real stress_error = diff.norm<L_inf>();
			ASSERT_NEAR(stress_error, 0., 1e-13);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElasticLinearAnisotropic<2>>::testEnergyDensity() {
			Matrix<Real> sigma = {{1, 2}, {2, 4}};
			Matrix<Real> eps = {{1, 0}, {0, 1}};
			Real epot = 0;
			Real solution = 2.5;
			this->computePotentialEnergyOnQuad(eps, sigma, epot);
			ASSERT_NEAR(epot, solution, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElasticLinearAnisotropic<2>>::testComputeTangentModuli() {

			// Note: for this test material and canonical basis coincide
			Matrix<Real> C = {
			{1.0, 0.3, 0.4},
			{0.3, 2.0, 0.1},
			{0.4, 0.1, 1.5},
			};

			setParamNoUpdate("C11", C(0,0));
			setParamNoUpdate("C12", C(0,1));
			setParamNoUpdate("C13", C(0,2));
			setParamNoUpdate("C22", C(1,1));
			setParamNoUpdate("C23", C(1,2));
			setParamNoUpdate("C33", C(2,2));

			// set internal Cijkl matrix expressed in the canonical frame of reference
			this->updateInternalParameters();

			Matrix<Real> tangent(3,3);
			this->computeTangentModuliOnQuad(tangent);

			Real tangent_error = (tangent - C).norm<L_2>();
			ASSERT_NEAR(tangent_error, 0, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElasticLinearAnisotropic<2>>::testPushWaveSpeed() {

			// Note: for this test material and canonical basis coincide
			Matrix<Real> C = {
			{1.0, 0.3, 0.4},
			{0.3, 2.0, 0.1},
			{0.4, 0.1, 1.5},
			};

			Real rho = 2.7;

			setParamNoUpdate("C11", C(0,0));
			setParamNoUpdate("C12", C(0,1));
			setParamNoUpdate("C13", C(0,2));
			setParamNoUpdate("C22", C(1,1));
			setParamNoUpdate("C23", C(1,2));
			setParamNoUpdate("C33", C(2,2));
			setParamNoUpdate("rho", rho);

			// set internal Cijkl matrix expressed in the canonical frame of reference
			this->updateInternalParameters();

			Vector<Real> eig_expected(3);
			C.eig(eig_expected);

			auto celerity_expected = std::sqrt(eig_expected(0)/rho);

			auto celerity = this->getCelerity(Element());

			ASSERT_NEAR(celerity_expected, celerity, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElasticLinearAnisotropic<2>>::testShearWaveSpeed() {

			AKANTU_DEBUG_WARNING("WARNING: Shear wave speed for 2D anisotropic materials is not present in akantu."
			"Celerity is used instead to get the stable time step (see testPushWaveSpeed).");
			ASSERT_NEAR(0, 0, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElasticLinearAnisotropic<3>>::testComputeStress() {
			UInt Dim = 3;
			Matrix<Real> C = {
			{1.0, 0.3, 0.4, 0.3, 0.2, 0.1},
			{0.3, 2.0, 0.1, 0.2, 0.3, 0.2},
			{0.4, 0.1, 1.5, 0.1, 0.4, 0.3},
			{0.3, 0.2, 0.1, 2.4, 0.1, 0.4},
			{0.2, 0.3, 0.4, 0.1, 0.9, 0.1},
			{0.1, 0.2, 0.3, 0.4, 0.1, 1.2},
			};

			setParamNoUpdate("C11", C(0,0));
			setParamNoUpdate("C12", C(0,1));
			setParamNoUpdate("C13", C(0,2));
			setParamNoUpdate("C14", C(0,3));
			setParamNoUpdate("C15", C(0,4));
			setParamNoUpdate("C16", C(0,5));
			setParamNoUpdate("C22", C(1,1));
			setParamNoUpdate("C23", C(1,2));
			setParamNoUpdate("C24", C(1,3));
			setParamNoUpdate("C25", C(1,4));
			setParamNoUpdate("C26", C(1,5));
			setParamNoUpdate("C33", C(2,2));
			setParamNoUpdate("C34", C(2,3));
			setParamNoUpdate("C35", C(2,4));
			setParamNoUpdate("C36", C(2,5));
			setParamNoUpdate("C44", C(3,3));
			setParamNoUpdate("C45", C(3,4));
			setParamNoUpdate("C46", C(3,5));
			setParamNoUpdate("C55", C(4,4));
			setParamNoUpdate("C56", C(4,5));
			setParamNoUpdate("C66", C(5,5));

			// material frame of reference is rotate by rotation_matrix starting from
			// canonical basis
			Matrix<Real> rotation_matrix = getRandomRotation3d();

			// canonical basis as expressed in the material frame of reference, as
			// required by MaterialElasticLinearAnisotropic class (it is simply given by
			// the columns of the rotation_matrix; the lines give the material basis
			// expressed in the canonical frame of reference)
			Vector<Real> v1(Dim);
			Vector<Real> v2(Dim);
			Vector<Real> v3(Dim);
			for (UInt i = 0; i < Dim; ++i) {
			v1[i] = rotation_matrix(i, 0);
			v2[i] = rotation_matrix(i, 1);
			v3[i] = rotation_matrix(i, 2);
			}

			setParamNoUpdate("n1", v1.normalize());
			setParamNoUpdate("n2", v2.normalize());
			setParamNoUpdate("n3", v3.normalize());

			// set internal Cijkl matrix expressed in the canonical frame of reference
			this->updateInternalParameters();

			// gradient in material frame of reference
			auto grad_u = this->getDeviatoricStrain(2.) + this->getHydrostaticStrain(1.);

			// gradient in canonical basis (we need to rotate back to the canonical
			// basis)
			auto grad_u_rot = this->reverseRotation(grad_u, rotation_matrix);

			// stress in the canonical basis
			Matrix<Real> sigma_rot(3, 3);
			this->computeStressOnQuad(grad_u_rot, sigma_rot);

			// stress in the material reference (we need to apply the rotation)
			auto sigma = this->applyRotation(sigma_rot, rotation_matrix);

			// epsilon is computed directly in the material frame of reference
			Matrix<Real> epsilon = 0.5 * ( grad_u + grad_u.transpose() );
			Vector<Real> epsilon_voigt(6);
			epsilon_voigt(0) = epsilon(0,0);
			epsilon_voigt(1) = epsilon(1,1);
			epsilon_voigt(2) = epsilon(2,2);
			epsilon_voigt(3) = 2 * epsilon(1,2);
			epsilon_voigt(4) = 2 * epsilon(0,2);
			epsilon_voigt(5) = 2 * epsilon(0,1);

			// sigma_expected is computed directly in the material frame of reference
			Vector<Real> sigma_voigt = C * epsilon_voigt;
			Matrix<Real> sigma_expected(Dim, Dim);
			sigma_expected(0,0) = sigma_voigt(0);
			sigma_expected(1,1) = sigma_voigt(1);
			sigma_expected(2,2) = sigma_voigt(2);
			sigma_expected(1,2) = sigma_expected(2,1) = sigma_voigt(3);
			sigma_expected(0,2) = sigma_expected(2,0) = sigma_voigt(4);
			sigma_expected(0,1) = sigma_expected(1,0) = sigma_voigt(5);

			// sigmas are checked in the material frame of reference
			auto diff = sigma - sigma_expected;
			Real stress_error = diff.norm<L_inf>();
			ASSERT_NEAR(stress_error, 0., 1e-13);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElasticLinearAnisotropic<3>>::testEnergyDensity() {
			Matrix<Real> sigma = {{1, 2, 3}, {2, 4, 5}, {3, 5, 6}};
			Matrix<Real> eps = {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}};
			Real epot = 0;
			Real solution = 5.5;
			this->computePotentialEnergyOnQuad(eps, sigma, epot);
			ASSERT_NEAR(epot, solution, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElasticLinearAnisotropic<3>>::testComputeTangentModuli() {

			// Note: for this test material and canonical basis coincide
			Matrix<Real> C = {
			{1.0, 0.3, 0.4, 0.3, 0.2, 0.1},
			{0.3, 2.0, 0.1, 0.2, 0.3, 0.2},
			{0.4, 0.1, 1.5, 0.1, 0.4, 0.3},
			{0.3, 0.2, 0.1, 2.4, 0.1, 0.4},
			{0.2, 0.3, 0.4, 0.1, 0.9, 0.1},
			{0.1, 0.2, 0.3, 0.4, 0.1, 1.2},
			};

			setParamNoUpdate("C11", C(0,0));
			setParamNoUpdate("C12", C(0,1));
			setParamNoUpdate("C13", C(0,2));
			setParamNoUpdate("C14", C(0,3));
			setParamNoUpdate("C15", C(0,4));
			setParamNoUpdate("C16", C(0,5));
			setParamNoUpdate("C22", C(1,1));
			setParamNoUpdate("C23", C(1,2));
			setParamNoUpdate("C24", C(1,3));
			setParamNoUpdate("C25", C(1,4));
			setParamNoUpdate("C26", C(1,5));
			setParamNoUpdate("C33", C(2,2));
			setParamNoUpdate("C34", C(2,3));
			setParamNoUpdate("C35", C(2,4));
			setParamNoUpdate("C36", C(2,5));
			setParamNoUpdate("C44", C(3,3));
			setParamNoUpdate("C45", C(3,4));
			setParamNoUpdate("C46", C(3,5));
			setParamNoUpdate("C55", C(4,4));
			setParamNoUpdate("C56", C(4,5));
			setParamNoUpdate("C66", C(5,5));

			// set internal Cijkl matrix expressed in the canonical frame of reference
			this->updateInternalParameters();

			Matrix<Real> tangent(6,6);
			this->computeTangentModuliOnQuad(tangent);

			Real tangent_error = (tangent - C).norm<L_2>();
			ASSERT_NEAR(tangent_error, 0, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElasticLinearAnisotropic<3>>::testPushWaveSpeed() {

			// Note: for this test material and canonical basis coincide
			Matrix<Real> C = {
			{1.0, 0.3, 0.4, 0.3, 0.2, 0.1},
			{0.3, 2.0, 0.1, 0.2, 0.3, 0.2},
			{0.4, 0.1, 1.5, 0.1, 0.4, 0.3},
			{0.3, 0.2, 0.1, 2.4, 0.1, 0.4},
			{0.2, 0.3, 0.4, 0.1, 0.9, 0.1},
			{0.1, 0.2, 0.3, 0.4, 0.1, 1.2},
			};
			Real rho = 2.9;

			setParamNoUpdate("C11", C(0,0));
			setParamNoUpdate("C12", C(0,1));
			setParamNoUpdate("C13", C(0,2));
			setParamNoUpdate("C14", C(0,3));
			setParamNoUpdate("C15", C(0,4));
			setParamNoUpdate("C16", C(0,5));
			setParamNoUpdate("C22", C(1,1));
			setParamNoUpdate("C23", C(1,2));
			setParamNoUpdate("C24", C(1,3));
			setParamNoUpdate("C25", C(1,4));
			setParamNoUpdate("C26", C(1,5));
			setParamNoUpdate("C33", C(2,2));
			setParamNoUpdate("C34", C(2,3));
			setParamNoUpdate("C35", C(2,4));
			setParamNoUpdate("C36", C(2,5));
			setParamNoUpdate("C44", C(3,3));
			setParamNoUpdate("C45", C(3,4));
			setParamNoUpdate("C46", C(3,5));
			setParamNoUpdate("C55", C(4,4));
			setParamNoUpdate("C56", C(4,5));
			setParamNoUpdate("C66", C(5,5));
			setParamNoUpdate("rho", rho);

			// set internal Cijkl matrix expressed in the canonical frame of reference
			this->updateInternalParameters();

			Vector<Real> eig_expected(6);
			C.eig(eig_expected);

			auto celerity_expected = std::sqrt(eig_expected(0)/rho);

			auto celerity = this->getCelerity(Element());

			ASSERT_NEAR(celerity_expected, celerity, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			template <> void FriendMaterial<MaterialElasticLinearAnisotropic<3>>::testShearWaveSpeed() {

			AKANTU_DEBUG_WARNING("WARNING: Shear wave speed for 3D anisotropic materials is not present in akantu."
			"Celerity is used instead to get the stable time step (see testPushWaveSpeed).");
			ASSERT_NEAR(0, 0, 1e-14);
			}

			/* -------------------------------------------------------------------------- */
			namespace {

			template <typename T>
			class TestElasticMaterialFixture : public ::TestMaterialFixture<T> {};

			TYPED_TEST_CASE(TestElasticMaterialFixture, types);

			TYPED_TEST(TestElasticMaterialFixture, ElasticComputeStress) {
			this->material->testComputeStress();
			}

			TYPED_TEST(TestElasticMaterialFixture, ElasticEnergyDensity) {
			this->material->testEnergyDensity();
			}

			TYPED_TEST(TestElasticMaterialFixture, ElasticComputeTangentModuli) {
			this->material->testComputeTangentModuli();
			}

			TYPED_TEST(TestElasticMaterialFixture, ElasticComputePushWaveSpeed) {
			this->material->testPushWaveSpeed();
			}

			TYPED_TEST(TestElasticMaterialFixture, ElasticComputeShearWaveSpeed) {
			this->material->testShearWaveSpeed();
			}
			} // namespace

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rAKA/test/test_model/test_solid_mechanics_model/test_materials/test_elastic_materials.cc
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