threed-beam-fea/tests/beam_element_tests.cpp

// Copyright 2015. All rights reserved.
//
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// modification, are permitted provided that the following conditions are met:
//
// * Redistributions of source code must retain the above copyright notice,
//   this list of conditions and the following disclaimer.
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//   this list of conditions and the following disclaimer in the documentation
//   and/or other materials provided with the distribution.
//
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
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// Author: ryan.latture@gmail.com (Ryan Latture)

#include "threed_beam_fea.h"
#include <gtest/gtest.h>

using namespace fea;

class beamFEATest : public testing::Test {
protected:
  beamFEATest()
      : JOB_L_BRACKET(), BCS_L_BRACKET(0), FORCES_L_BRACKET(0),
        JOB_CANTILEVER(), BCS_CANTILEVER(0), FORCES_CANTILEVER(0),
        assembleK3D(){};

  virtual void SetUp() {
    using namespace fea;

    // L-bracket setup
    std::vector<double> normal_vec = {0.0, 1.0, 0.0};
    Props PROPS1(10.0, 10.0, 10.0, 10.0, normal_vec);
    Props PROPS2(10.0, 1.0, 1.0, 10.0, normal_vec);

    std::vector<Node> NODES = {Node(0.0, 0.0, 0.0), Node(1.0, 0.0, 0.0),
                               Node(2.0, 0.0, 0.0), Node(2.0, 0.0, 1.0)};
    std::vector<Elem> ELEMS = {Elem(0, 1, PROPS1), Elem(1, 2, PROPS1),
                               Elem(2, 3, PROPS2)};

    JOB_L_BRACKET = Job(NODES, ELEMS);

    BC bc1(0, 0, 0.0);
    BC bc2(0, 1, 0.0);
    BC bc3(0, 2, 0.0);
    BC bc4(0, 3, 0.0);
    BC bc5(0, 4, 0.0);
    BC bc6(0, 5, 0.0);
    BC bc7(3, 1, 0.5);

    BCS_L_BRACKET = {bc1, bc2, bc3, bc4, bc5, bc6, bc7};

    // cantilever setup
    std::vector<double> normal_cantilever = {0.0, 0.0, 1.0};
    Props props_cantilever(1.0, 1.0, 1.0, 1.0, normal_cantilever);
    std::vector<Node> nodes_cantilever = {Node(0.0, 0.0, 0.0),
                                          Node(1.0, 0.0, 0.0)};

    std::vector<Elem> elems_cantilever = {Elem(0, 1, props_cantilever)};

    BCS_CANTILEVER = {BC(0, 0, 0.0), BC(0, 1, 0.0), BC(0, 2, 0.0),
                      BC(0, 3, 0.0), BC(0, 4, 0.0), BC(0, 5, 0.0)};

    FORCES_CANTILEVER = {Force(1, 1, 0.1)};

    JOB_CANTILEVER = Job(nodes_cantilever, elems_cantilever);
  }

  Job JOB_L_BRACKET;
  std::vector<BC> BCS_L_BRACKET;
  std::vector<Force> FORCES_L_BRACKET;

  Job JOB_CANTILEVER;
  std::vector<BC> BCS_CANTILEVER;
  std::vector<Force> FORCES_CANTILEVER;

  GlobalStiffAssembler assembleK3D;

public:
  EIGEN_MAKE_ALIGNED_OPERATOR_NEW;
};

TEST_F(beamFEATest, TransformsLocalToGlobalCoords) {
  // test that identity matrix is recovered if local axes == global axes
  Eigen::Vector3d nx(1.0, 0.0, 0.0);
  Eigen::Vector3d nz(0.0, 0.0, 1.0);
  const Eigen::Vector3d ny = nz.cross(nx).normalized();
  RotationMatrix r;
  r.row(0) = nx;
  r.row(1) = ny;
  r.row(2) = nz;

  assembleK3D.calcAelem(r);

  LocalMatrix Aelem = assembleK3D.getAelem();

  LocalMatrix expected;
  expected.setIdentity();

  for (size_t i = 0; i < Aelem.rows(); ++i) {
    EXPECT_DOUBLE_EQ(expected(i), Aelem(i));
  }
}

TEST_F(beamFEATest, AssemblesElementalStiffness) {

  assembleK3D.calcKelem(0, JOB_L_BRACKET);
  LocalMatrix Kelem = assembleK3D.getKelem();

  std::vector<std::vector<double>> expected = {
      {10., 0., 0., 0., 0., 0., -10., 0., 0., 0., 0., 0.},
      {0., 120., 0., 0., 0., 60., 0., -120., 0., 0., 0., 60.},
      {0., 0., 120., 0., -60., 0., 0., 0., -120., 0., -60., 0.},
      {0., 0., 0., 10., 0., 0., 0., 0., 0., -10., 0., 0.},
      {0., 0., -60., 0., 40., 0., 0., 0., 60., 0., 20., 0.},
      {0., 60., 0., 0., 0., 40., 0., -60., 0., 0., 0., 20.},
      {-10., 0., 0., 0., 0., 0., 10., 0., 0., 0., 0., 0.},
      {0., -120., 0., 0., 0., -60., 0., 120., 0., 0., 0., -60.},
      {0., 0., -120., 0., 60., 0., 0., 0., 120., 0., 60., 0.},
      {0., 0., 0., -10., 0., 0., 0., 0., 0., 10., 0., 0.},
      {0., 0., -60., 0., 20., 0., 0., 0., 60., 0., 40., 0.},
      {0., 60., 0., 0., 0., 20., 0., -60., 0., 0., 0., 40.}};

  for (size_t i = 0; i < expected.size(); i++) {
    for (size_t j = 0; j < expected[i].size(); j++) {
      EXPECT_DOUBLE_EQ(expected[i][j], Kelem(i, j));
    }
  }
}

TEST_F(beamFEATest, AssemblesGlobalStiffness) {

  unsigned int dofs_per_elem = 6;

  // calculate size of global stiffness matrix and force vector
  size_t size =
      dofs_per_elem * JOB_L_BRACKET.nodes.size() + FORCES_L_BRACKET.size();

  // construct global stiffness matrix and force vector
  SparseMat Kg(size, size);

  std::vector<Tie> ties;

  assembleK3D(Kg, JOB_L_BRACKET, ties);

  GlobalStiffMatrix KgDense(Kg);

  std::vector<std::vector<double>> expected = {
      {10., 0., 0., 0., 0., 0., -10., 0., 0., 0., 0., 0.,
       0.,  0., 0., 0., 0., 0., 0.,   0., 0., 0., 0., 0.},
      {0., 120., 0., 0., 0., 60., 0., -120., 0., 0., 0., 60.,
       0., 0.,   0., 0., 0., 0.,  0., 0.,    0., 0., 0., 0.},
      {0., 0., 120., 0., -60., 0., 0., 0., -120., 0., -60., 0.,
       0., 0., 0.,   0., 0.,   0., 0., 0., 0.,    0., 0.,   0.},
      {0., 0., 0., 10., 0., 0., 0., 0., 0., -10., 0., 0.,
       0., 0., 0., 0.,  0., 0., 0., 0., 0., 0.,   0., 0.},
      {0., 0., -60., 0., 40., 0., 0., 0., 60., 0., 20., 0.,
       0., 0., 0.,   0., 0.,  0., 0., 0., 0.,  0., 0.,  0.},
      {0., 60., 0., 0., 0., 40., 0., -60., 0., 0., 0., 20.,
       0., 0.,  0., 0., 0., 0.,  0., 0.,   0., 0., 0., 0.},
      {-10., 0., 0., 0., 0., 0., 20., 0., 0., 0., 0., 0.,
       -10., 0., 0., 0., 0., 0., 0.,  0., 0., 0., 0., 0.},
      {0., -120., 0., 0., 0., -60., 0., 240., 0., 0., 0., 0.,
       0., -120., 0., 0., 0., 60.,  0., 0.,   0., 0., 0., 0.},
      {0., 0., -120., 0., 60.,  0., 0., 0., 240., 0., 0., 0.,
       0., 0., -120., 0., -60., 0., 0., 0., 0.,   0., 0., 0.},
      {0., 0., 0., -10., 0., 0., 0., 0., 0., 20., 0., 0.,
       0., 0., 0., -10., 0., 0., 0., 0., 0., 0.,  0., 0.},
      {0., 0., -60., 0., 20., 0., 0., 0., 0., 0., 80., 0.,
       0., 0., 60.,  0., 20., 0., 0., 0., 0., 0., 0.,  0.},
      {0., 60.,  0., 0., 0., 20., 0., 0., 0., 0., 0., 80.,
       0., -60., 0., 0., 0., 20., 0., 0., 0., 0., 0., 0.},
      {0.,  0., 0., 0., 0., 0., -10., 0., 0., 0., 0., 0.,
       22., 0., 0., 0., 6., 0., -12., 0., 0., 0., 6., 0.},
      {0., 0.,   0., 0.,  0., 0.,   0., -120., 0., 0.,  0., -60.,
       0., 132., 0., -6., 0., -60., 0., -12.,  0., -6., 0., 0.},
      {0., 0., 0.,   0., 0.,  0., 0., 0., -120., 0., 60., 0.,
       0., 0., 130., 0., 60., 0., 0., 0., -10.,  0., 0.,  0.},
      {0., 0.,  0., 0.,  0., 0., 0., 0., 0., -10., 0., 0.,
       0., -6., 0., 14., 0., 0., 0., 6., 0., 2.,   0., 0.},
      {0., 0., 0.,  0., 0.,  0., 0.,  0., -60., 0., 20., 0.,
       6., 0., 60., 0., 44., 0., -6., 0., 0.,   0., 2.,  0.},
      {0., 0.,   0., 0., 0., 0.,  0., 60., 0., 0., 0., 20.,
       0., -60., 0., 0., 0., 50., 0., 0.,  0., 0., 0., -10.},
      {0.,   0., 0., 0., 0.,  0., 0.,  0., 0., 0., 0.,  0.,
       -12., 0., 0., 0., -6., 0., 12., 0., 0., 0., -6., 0.},
      {0., 0.,   0., 0., 0., 0., 0., 0.,  0., 0., 0., 0.,
       0., -12., 0., 6., 0., 0., 0., 12., 0., 6., 0., 0.},
      {0., 0., 0.,   0., 0., 0., 0., 0., 0.,  0., 0., 0.,
       0., 0., -10., 0., 0., 0., 0., 0., 10., 0., 0., 0.},
      {0., 0.,  0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
       0., -6., 0., 2., 0., 0., 0., 6., 0., 4., 0., 0.},
      {0., 0., 0., 0., 0., 0., 0.,  0., 0., 0., 0., 0.,
       6., 0., 0., 0., 2., 0., -6., 0., 0., 0., 4., 0.},
      {0., 0., 0., 0., 0., 0.,   0., 0., 0., 0., 0., 0.,
       0., 0., 0., 0., 0., -10., 0., 0., 0., 0., 0., 10.}};

  for (size_t i = 0; i < expected.size(); i++) {
    for (size_t j = 0; j < expected[i].size(); j++) {
      EXPECT_DOUBLE_EQ(expected[i][j], KgDense(i, j));
    }
  }
}

TEST_F(beamFEATest, CorrectNodalDisplacementsNoTies) {
  std::vector<Tie> ties;
  std::vector<Equation> equations;
  Summary summary = solve(JOB_L_BRACKET, BCS_L_BRACKET, FORCES_L_BRACKET, ties,
                          equations, Options());

  // the first 4 rows check nodal displacements
  // the last row is associated with the reaction
  // forces due to enforcing the BCs.
  std::vector<std::vector<double>> expected = {
      {0., 0., 0., 0., 0., 0.},
      {0., 0.0520833333333333, 0., -0.0625, 0., 0.09375},
      {0., 0.16666666666666666, 0., -0.125, 0., 0.125},
      {0., 0.5, 0., -0.4375, 0., 0.125},
      {0., 0.625, 0., -0.625, 0., 1.25, -0.625}};

  for (size_t i = 0; i < summary.nodal_displacements.size(); ++i) {
    for (size_t j = 0; j < summary.nodal_displacements[i].size(); ++j) {
      EXPECT_NEAR(expected[i][j], summary.nodal_displacements[i][j], 1.e-10);
    }
  }
}

// In this test I create the same mesh as above with 1 exception
// there are redundant nodes at (1,0,0) that I tie with extremely
// stiff linear and rotational springs. If the tie constraints are
// functioning properly, the ties should behave the same as having
// rigid nodal joints, at least to several decimal places.
TEST_F(beamFEATest, CorrectNodalDisplacementsWithStiffTies) {
  std::vector<double> normal_vec = {0.0, 1.0, 0.0};
  Props props1(10.0, 10.0, 10.0, 10.0, normal_vec);
  Props props2(10.0, 1.0, 1.0, 10.0, normal_vec);

  std::vector<Node> nodes = {Node(0.0, 0.0, 0.0), Node(1.0, 0.0, 0.0),
                             Node(1.0, 0.0, 0.0), Node(2.0, 0.0, 0.0),
                             Node(2.0, 0.0, 1.0)};
  std::vector<Elem> elems = {Elem(0, 1, props1), Elem(2, 3, props1),
                             Elem(3, 4, props2)};

  Job job_tie(nodes, elems);

  BC bc1(0, 0, 0.0);
  BC bc2(0, 1, 0.0);
  BC bc3(0, 2, 0.0);
  BC bc4(0, 3, 0.0);
  BC bc5(0, 4, 0.0);
  BC bc6(0, 5, 0.0);
  BC bc7(4, 1, 0.5);

  std::vector<BC> bcs = {bc1, bc2, bc3, bc4, bc5, bc6, bc7};

  std::vector<Tie> ties = {Tie(1, 2, 1.e8, 1.e8)};

  std::vector<Force> forces;

  std::vector<Equation> equations;

  Summary summary = solve(job_tie, bcs, forces, ties, equations, Options());

  // The verification program used to generate expected values outputs data as
  // floats
  std::vector<std::vector<double>> expected = {
      {0., 0., 0., 0., 0., 0.},
      {0., 0.0520833333333333, 0., -0.0625, 0., 0.09375},
      {0., 0.0520833333333333, 0., -0.0625, 0., 0.09375},
      {0., 0.16666666666666666, 0., -0.125, 0., 0.125},
      {0., 0.5, 0., -0.4375, 0., 0.125},
      {0., 0.625, 0., -0.625, 0., 1.25, -0.625}};

  for (size_t i = 0; i < summary.nodal_displacements.size(); ++i) {
    for (size_t j = 0; j < summary.nodal_displacements[i].size(); ++j) {
      EXPECT_NEAR(expected[i][j], summary.nodal_displacements[i][j], 1.e-7);
    }
  }
}

TEST_F(beamFEATest, CorrectDisplacementWithEquationsCantileverBeam) {
  std::vector<BC> bcs = {BC(0, 0, 0.1), BC(0, 1, 0.0), BC(0, 2, 0.0),
                         BC(0, 3, 0.0), BC(0, 4, 0.0), BC(0, 5, 0.0)};

  std::vector<Tie> ties;
  std::vector<Force> forces;

  Equation eqn;
  eqn.terms.push_back(Equation::Term(0, 0, 1));
  eqn.terms.push_back(Equation::Term(1, 0, 1));
  std::vector<Equation> equations = {eqn};

  Summary summary =
      solve(JOB_CANTILEVER, bcs, forces, ties, equations, Options());

  std::vector<std::vector<double>> expected = {{0.1, 0., 0., 0., 0., 0.},
                                               {-0.1, 0., 0., 0., 0., 0.}};

  for (size_t i = 0; i < summary.nodal_displacements.size(); ++i) {
    for (size_t j = 0; j < summary.nodal_displacements[i].size(); ++j)
      EXPECT_DOUBLE_EQ(expected[i][j], summary.nodal_displacements[i][j]);
  }
}

// This is simple cantilever beam with a point load at the
// free end. This checks the end displacements match the analytical results.
TEST_F(beamFEATest, CorrectTipDisplacementCantileverBeam) {

  std::vector<Tie> ties;
  std::vector<Equation> equations;

  Summary summary = solve(JOB_CANTILEVER, BCS_CANTILEVER, FORCES_CANTILEVER,
                          ties, equations, Options());

  // the first row checks nodal displacements of fixed node are zero
  // the second row checks the analytical result for tip displacement
  // for the given load in the y-direction of 0.01 yields the correct
  // displacement and rotation
  std::vector<std::vector<double>> expected = {
      {0., 0., 0., 0., 0., 0.}, {0., 0.033333333333333333, 0., 0., 0., 0.05}};

  for (size_t i = 0; i < summary.nodal_displacements.size(); ++i) {
    for (size_t j = 0; j < summary.nodal_displacements[i].size(); ++j)
      EXPECT_DOUBLE_EQ(expected[i][j], summary.nodal_displacements[i][j]);
  }
}

// This test displaces a cantilever beam axially and
// transverse to the beam axis. Nodal forces are
// compared to the analytical result.
TEST_F(beamFEATest, CorrectTipForcesCantileverBeam) {

  std::vector<Tie> ties;
  std::vector<Equation> equations;
  std::vector<Force> forces;
  std::vector<BC> bcs = BCS_CANTILEVER;
  bcs.push_back(BC(1, 0, 0.1));
  bcs.push_back(BC(1, 1, 0.1));

  Options opts;

  Summary summary = solve(JOB_CANTILEVER, bcs, forces, ties, equations, opts);

  std::vector<std::vector<double>> expected = {{-0.1, -0.3, 0., 0., 0., -0.3},
                                               {0.1, 0.3, 0., 0., 0., 0.}};

  for (size_t i = 0; i < summary.nodal_forces.size(); ++i) {
    for (size_t j = 0; j < summary.nodal_forces[i].size(); ++j)
      EXPECT_DOUBLE_EQ(expected[i][j], summary.nodal_forces[i][j]);
  }
}

// This tests the tie constraints correctly deform.
// I have extremely stiff elements and apply a
// displacement to the end node. It is expected
// that the tie will accommodate all the deformation.
TEST_F(beamFEATest, CorrectDisplacementWeakTies) {

  std::vector<double> normal_vec = {0.0, 1.0, 0.0};

  Props props(1.e9, 1.e9, 1.e9, 1.e9, normal_vec);

  std::vector<Node> nodes = {Node(0.0, 0.0, 0.0), Node(1.0, 0.0, 0.0),
                             Node(1.0, 0.0, 0.0), Node(2.0, 0.0, 0.0)};

  std::vector<Elem> elems = {Elem(0, 1, props), Elem(2, 3, props)};

  Job job_tie(nodes, elems);

  BC bc1(0, 0, 0.0);
  BC bc2(0, 1, 0.0);
  BC bc3(0, 2, 0.0);
  BC bc4(0, 3, 0.0);
  BC bc5(0, 4, 0.0);
  BC bc6(0, 5, 0.0);
  BC bc7(3, 0, 0.5);

  std::vector<BC> bcs = {bc1, bc2, bc3, bc4, bc5, bc6, bc7};

  std::vector<Tie> ties = {Tie(1, 2, 0.01, 0.01)};

  std::vector<Force> forces;
  std::vector<Equation> equations;

  Options opts;
  opts.epsilon = 1e-10;

  Summary summary = solve(job_tie, bcs, forces, ties, equations, opts);

  std::vector<std::vector<double>> expected = {{0.0, 0.0, 0.0, 0.0, 0.0, 0.0},
                                               {0.0, 0.0, 0.0, 0.0, 0.0, 0.0},
                                               {0.5, 0.0, 0.0, 0.0, 0.0, 0.0},
                                               {0.5, 0.0, 0.0, 0.0, 0.0, 0.0}};

  for (size_t i = 0; i < summary.nodal_displacements.size(); ++i) {
    for (size_t j = 0; j < summary.nodal_displacements[i].size(); ++j) {
      EXPECT_NEAR(expected[i][j], summary.nodal_displacements[i][j], 1e-10);
    }
  }
}

TEST_F(beamFEATest, CorrectForcesWeakTies) {

  std::vector<double> normal_vec = {0.0, 1.0, 0.0};

  Props props(1.e9, 1.e9, 1.e9, 1.e9, normal_vec);

  std::vector<Node> nodes = {Node(0.0, 0.0, 0.0), Node(1.0, 0.0, 0.0),
                             Node(1.0, 0.0, 0.0), Node(2.0, 0.0, 0.0)};

  std::vector<Elem> elems = {Elem(0, 1, props), Elem(2, 3, props)};

  Job job_tie(nodes, elems);

  BC bc1(0, 0, 0.0);
  BC bc2(0, 1, 0.0);
  BC bc3(0, 2, 0.0);
  BC bc4(0, 3, 0.0);
  BC bc5(0, 4, 0.0);
  BC bc6(0, 5, 0.0);
  BC bc7(3, 0, 0.5);
  BC bc8(2, 3, 0.5);

  std::vector<BC> bcs = {bc1, bc2, bc3, bc4, bc5, bc6, bc7, bc8};

  std::vector<Tie> ties = {Tie(1, 2, 0.01, 0.01)};

  std::vector<Force> forces;
  std::vector<Equation> equations;

  Options opts;
  opts.epsilon = 1e-10;

  Summary summary = solve(job_tie, bcs, forces, ties, equations, opts);

  std::vector<std::vector<double>> expected = {
      {0.005, 0.0, 0.0, 0.005, 0.0, 0.0}};

  for (size_t i = 0; i < summary.tie_forces.size(); ++i) {
    for (size_t j = 0; j < summary.tie_forces[i].size(); ++j) {
      EXPECT_NEAR(expected[i][j], summary.tie_forces[i][j], 1e-13);
    }
  }
}