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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>>::testCelerity() {
			Real E = 3., rho = 2.;
			setParam("E", E);
			setParam("rho", rho);

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

			/* -------------------------------------------------------------------------- */
			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);

			Real bulk_modulus_K = E / 3. / (1 - 2. * nu);
			Real shear_modulus_mu = 0.5 * E / (1 + nu);

			Matrix<Real> rotation_matrix = getRandomRotation2d();

			auto grad_u = this->getComposedStrain(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> strain = 0.5 * (grad_u + grad_u.transpose());
			Matrix<Real> deviatoric_strain = strain - 1. / 3. * strain.trace() * identity;

			Matrix<Real> sigma_expected = 2 * shear_modulus_mu * deviatoric_strain +
			(sigma_th + 2. * bulk_modulus_K) * 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>>::testCelerity() {
			Real E = 1.;
			Real nu = .3;
			Real rho = 2;
			setParam("E", E);
			setParam("nu", nu);
			setParam("rho", rho);
			auto push_wave_speed = this->getPushWaveSpeed(Element());
			auto celerity = this->getCelerity(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(push_wave_speed, sol, 1e-14);
			ASSERT_NEAR(celerity, sol, 1e-14);

			auto shear_wave_speed = this->getShearWaveSpeed(Element());

			sol = std::sqrt(mu / rho);

			ASSERT_NEAR(shear_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);

			Real bulk_modulus_K = E / 3. / (1 - 2. * nu);
			Real shear_modulus_mu = 0.5 * E / (1 + nu);

			Matrix<Real> rotation_matrix = getRandomRotation3d();

			auto grad_u = this->getComposedStrain(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> strain = 0.5 * (grad_u + grad_u.transpose());
			Matrix<Real> deviatoric_strain = strain - 1. / 3. * strain.trace() * identity;

			Matrix<Real> sigma_expected = 2 * shear_modulus_mu * deviatoric_strain +
			(sigma_th + 3. * bulk_modulus_K) * 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>>::testCelerity() {
			Real E = 1.;
			Real nu = .3;
			Real rho = 2;
			setParam("E", E);
			setParam("nu", nu);
			setParam("rho", rho);

			auto push_wave_speed = this->getPushWaveSpeed(Element());
			auto celerity = this->getCelerity(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(push_wave_speed, sol, 1e-14);
			ASSERT_NEAR(celerity, sol, 1e-14);

			auto shear_wave_speed = this->getShearWaveSpeed(Element());

			sol = std::sqrt(mu / rho);

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

			/* -------------------------------------------------------------------------- */
			template <>
			void FriendMaterial<MaterialElasticOrthotropic<2>>::testComputeStress() {
			UInt Dim = 2;

			this->E1 = 1.;
			this->E2 = 2.;
			this->nu12 = 0.1;
			this->G12 = 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 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)
			*this->dir_vecs[0] = rotation_matrix(0);
			*this->dir_vecs[1] = rotation_matrix(1);

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

			// gradient in material frame of reference
			auto grad_u = this->getComposedStrain(2.).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(2 * Dim, 2 * Dim, 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.;

			*this->dir_vecs[0] = n1;
			*this->dir_vecs[1] = n2;

			this->E1 = E1;
			this->E2 = E2;
			this->nu12 = nu12;
			this->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>>::testCelerity() {

			// 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;

			*this->dir_vecs[0] = n1;
			*this->dir_vecs[1] = n2;
			this->E1 = E1;
			this->E2 = E2;
			this->nu12 = nu12;
			this->G12 = G12;
			this->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<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.;

			this->E1 = E1;
			this->E2 = E2;
			this->E3 = E3;
			this->nu12 = nu12;
			this->nu13 = nu13;
			this->nu23 = nu23;
			this->G12 = G12;
			this->G13 = G13;
			this->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)
			*this->dir_vecs[0] = rotation_matrix(0);
			*this->dir_vecs[1] = rotation_matrix(1);
			*this->dir_vecs[2] = rotation_matrix(2);

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

			// gradient in material frame of reference
			auto grad_u = this->getComposedStrain(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(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 - nu12 * nu21 - nu23 * nu32 - nu31 * nu13 -
			2 * nu21 * nu32 * nu13);

			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.;

			*this->dir_vecs[0] = n1;
			*this->dir_vecs[1] = n2;
			*this->dir_vecs[2] = n3;
			this->E1 = E1;
			this->E2 = E2;
			this->E3 = E3;
			this->nu12 = nu12;
			this->nu13 = nu13;
			this->nu23 = nu23;
			this->G12 = G12;
			this->G13 = G13;
			this->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 - nu12 * nu21 - nu23 * nu32 - nu31 * nu13 -
			2 * nu21 * nu32 * nu13);

			Matrix<Real> C_expected(2 * Dim, 2 * Dim, 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>>::testCelerity() {

			// 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;

			*this->dir_vecs[0] = n1;
			*this->dir_vecs[1] = n2;
			*this->dir_vecs[2] = n3;
			this->E1 = E1;
			this->E2 = E2;
			this->E3 = E3;
			this->nu12 = nu12;
			this->nu13 = nu13;
			this->nu23 = nu23;
			this->G12 = G12;
			this->G13 = G13;
			this->G23 = G23;
			this->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 - nu12 * nu21 - nu23 * nu32 - nu31 * nu13 -
			2 * nu21 * nu32 * nu13);

			Matrix<Real> C_expected(2 * Dim, 2 * Dim, 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<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},
			};

			for (auto i = 0u; i < C.rows(); ++i)
			for (auto j = 0u; j < C.cols(); ++j)
			this->Cprime(i, j) = C(i, j);

			// 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)
			*this->dir_vecs[0] = rotation_matrix(0);
			*this->dir_vecs[1] = rotation_matrix(1);

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

			// gradient in material frame of reference
			auto grad_u = this->getComposedStrain(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},
			};

			for (auto i = 0u; i < C.rows(); ++i)
			for (auto j = 0u; j < C.cols(); ++j)
			this->Cprime(i, j) = C(i, j);

			// 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>>::testCelerity() {

			// 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;

			for (auto i = 0u; i < C.rows(); ++i)
			for (auto j = 0u; j < C.cols(); ++j)
			this->Cprime(i, j) = C(i, j);

			this->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<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},
			};

			for (auto i = 0u; i < C.rows(); ++i)
			for (auto j = 0u; j < C.cols(); ++j)
			this->Cprime(i, j) = C(i, j);

			// 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)
			*this->dir_vecs[0] = rotation_matrix(0);
			*this->dir_vecs[1] = rotation_matrix(1);
			*this->dir_vecs[2] = rotation_matrix(2);

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

			// gradient in material frame of reference
			auto grad_u = this->getComposedStrain(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(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},
			};

			for (auto i = 0u; i < C.rows(); ++i)
			for (auto j = 0u; j < C.cols(); ++j)
			this->Cprime(i, j) = C(i, j);

			// 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>>::testCelerity() {

			// 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;

			for (auto i = 0u; i < C.rows(); ++i)
			for (auto j = 0u; j < C.cols(); ++j)
			this->Cprime(i, j) = C(i, j);

			this->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);
			}

			/* -------------------------------------------------------------------------- */

			namespace {

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

			TYPED_TEST_CASE(TestElasticMaterialFixture, types);

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

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

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

			TYPED_TEST(TestElasticMaterialFixture, ComputeCelerity) {
			this->material->testCelerity();
			}

			} // namespace

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