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<title>Computing Energy, Gradient, and Hessian - Physics-Based Simulation</title>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="preface.html">Preface</a></li><li class="chapter-item expanded affix "><li class="part-title">Simulation with Optimization</li><li class="chapter-item expanded "><a href="lec1-discrete_space_time.html"><strong aria-hidden="true">1.</strong> Discrete Space and Time</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec1.1-solid_rep.html"><strong aria-hidden="true">1.1.</strong> Representations of a Solid Geometry</a></li><li class="chapter-item expanded "><a href="lec1.2-newton_2nd_law.html"><strong aria-hidden="true">1.2.</strong> Newton's Second Law</a></li><li class="chapter-item expanded "><a href="lec1.3-time_integration.html"><strong aria-hidden="true">1.3.</strong> Time Integration</a></li><li class="chapter-item expanded "><a href="lec1.4-explicit_time_integration.html"><strong aria-hidden="true">1.4.</strong> Explicit Time Integration</a></li><li class="chapter-item expanded "><a href="lec1.5-implicit_time_integration.html"><strong aria-hidden="true">1.5.</strong> Implicit Time integration</a></li><li class="chapter-item expanded "><a href="lec1.6-summary.html"><strong aria-hidden="true">1.6.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec2-opt_framework.html"><strong aria-hidden="true">2.</strong> Optimization Framework</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec2.1-opt_time_integration.html"><strong aria-hidden="true">2.1.</strong> Optimization Time Integrator</a></li><li class="chapter-item expanded "><a href="lec2.2-dirichlet_BC.html"><strong aria-hidden="true">2.2.</strong> Dirichlet Boundary Conditions</a></li><li class="chapter-item expanded "><a href="lec2.3-contact.html"><strong aria-hidden="true">2.3.</strong> Contact</a></li><li class="chapter-item expanded "><a href="lec2.4-friction.html"><strong aria-hidden="true">2.4.</strong> Friction</a></li><li class="chapter-item expanded "><a href="lec2.5-summary.html"><strong aria-hidden="true">2.5.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec3-projected_Newton.html"><strong aria-hidden="true">3.</strong> Projected Newton</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec3.1-conv_issue_Newton.html"><strong aria-hidden="true">3.1.</strong> Convergence of Newton's Method</a></li><li class="chapter-item expanded "><a href="lec3.2-line_search.html"><strong aria-hidden="true">3.2.</strong> Line Search</a></li><li class="chapter-item expanded "><a href="lec3.3-grad_based_opt.html"><strong aria-hidden="true">3.3.</strong> Gradient-Based Optimization</a></li><li class="chapter-item expanded "><a href="lec3.4-summary.html"><strong aria-hidden="true">3.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec4-2d_mass_spring.html"><strong aria-hidden="true">4.</strong> Case Study: 2D Mass-Spring*</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec4.1-discretizations.html"><strong aria-hidden="true">4.1.</strong> Spatial and Temporal Discretizations</a></li><li class="chapter-item expanded "><a href="lec4.2-inertia.html"><strong aria-hidden="true">4.2.</strong> Inertia Term</a></li><li class="chapter-item expanded "><a href="lec4.3-mass_spring_energy.html"><strong aria-hidden="true">4.3.</strong> Mass-Spring Potential Energy</a></li><li class="chapter-item expanded "><a href="lec4.4-opt_time_integrator.html"><strong aria-hidden="true">4.4.</strong> Optimization Time Integrator</a></li><li class="chapter-item expanded "><a href="lec4.5-sim_with_vis.html"><strong aria-hidden="true">4.5.</strong> Simulation with Visualization</a></li><li class="chapter-item expanded "><a href="lec4.6-gpu_accel.html"><strong aria-hidden="true">4.6.</strong> GPU-Accelerated Simulation</a></li><li class="chapter-item expanded "><a href="lec4.6-summary.html"><strong aria-hidden="true">4.7.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><li class="part-title">Boundary Treatments</li><li class="chapter-item expanded "><a href="lec5-dirichlet_BC_solve.html"><strong aria-hidden="true">5.</strong> Dirichlet Boundary Conditions*</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec5.1-equality_constraints.html"><strong aria-hidden="true">5.1.</strong> Equality Constraint Formulation</a></li><li class="chapter-item expanded "><a href="lec5.2-DOF_elimin.html"><strong aria-hidden="true">5.2.</strong> DOF Elimination Method</a></li><li class="chapter-item expanded "><a href="lec5.3-hanging_square.html"><strong aria-hidden="true">5.3.</strong> Case Study: Hanging Sqaure*</a></li><li class="chapter-item expanded "><a href="lec5.4-summary.html"><strong aria-hidden="true">5.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec6-slip_DBC.html"><strong aria-hidden="true">6.</strong> Slip Dirichlet Boundary Conditions</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec6.1-axis_aligned.html"><strong aria-hidden="true">6.1.</strong> Axis-Aligned Slip DBC</a></li><li class="chapter-item expanded "><a href="lec6.2-change_of_vars.html"><strong aria-hidden="true">6.2.</strong> Change of Variables</a></li><li class="chapter-item expanded "><a href="lec6.3-general_slip_DBC.html"><strong aria-hidden="true">6.3.</strong> General Slip DBC</a></li><li class="chapter-item expanded "><a href="lec6.4-summary.html"><strong aria-hidden="true">6.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec7-dist_barrier.html"><strong aria-hidden="true">7.</strong> Distance Barrier for Nonpenetration</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec7.1-signed_dists.html"><strong aria-hidden="true">7.1.</strong> Signed Distances</a></li><li class="chapter-item expanded "><a href="lec7.2-dist_barrier_formulation.html"><strong aria-hidden="true">7.2.</strong> Distance Barrier</a></li><li class="chapter-item expanded "><a href="lec7.3-sol_accuracy.html"><strong aria-hidden="true">7.3.</strong> Solution Accuracy</a></li><li class="chapter-item expanded "><a href="lec7.4-summary.html"><strong aria-hidden="true">7.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec8-filter_line_search.html"><strong aria-hidden="true">8.</strong> Filter Line Search*</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec8.1-tunneling.html"><strong aria-hidden="true">8.1.</strong> Tunneling Issue</a></li><li class="chapter-item expanded "><a href="lec8.2-nonpenetration_traj.html"><strong aria-hidden="true">8.2.</strong> Penetration-free Trajectory</a></li><li class="chapter-item expanded "><a href="lec8.3-square_drop.html"><strong aria-hidden="true">8.3.</strong> Case Study: Square Drop*</a></li><li class="chapter-item expanded "><a href="lec8.4-summary.html"><strong aria-hidden="true">8.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec9-friction.html"><strong aria-hidden="true">9.</strong> Frictional Contact</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec9.1-smooth_fric.html"><strong aria-hidden="true">9.1.</strong> Smooth Dynamic-Static Transition</a></li><li class="chapter-item expanded "><a href="lec9.2-semi_imp_fric.html"><strong aria-hidden="true">9.2.</strong> Semi-Implicit Discretization</a></li><li class="chapter-item expanded "><a href="lec9.3-fixed_point_iter.html"><strong aria-hidden="true">9.3.</strong> Fixed-Point Iteration</a></li><li class="chapter-item expanded "><a href="lec9.4-summary.html"><strong aria-hidden="true">9.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec10-square_on_slope.html"><strong aria-hidden="true">10.</strong> Case Study: Square On Slope*</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec10.1-ground_to_slope.html"><strong aria-hidden="true">10.1.</strong> From Ground To Slope</a></li><li class="chapter-item expanded "><a href="lec10.2-slope_fric.html"><strong aria-hidden="true">10.2.</strong> Slope Friction</a></li><li class="chapter-item expanded "><a href="lec10.3-summary.html"><strong aria-hidden="true">10.3.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec11-mov_DBC.html"><strong aria-hidden="true">11.</strong> Moving Boundary Conditions*</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec11.1-penalty_method.html"><strong aria-hidden="true">11.1.</strong> Penalty Method</a></li><li class="chapter-item expanded "><a href="lec11.2-compress_square.html"><strong aria-hidden="true">11.2.</strong> Case Study: Compressing Square*</a></li><li class="chapter-item expanded "><a href="lec11.3-summary.html"><strong aria-hidden="true">11.3.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><li class="part-title">Hyperelasticity</li><li class="chapter-item expanded "><a href="lec12-kinematics.html"><strong aria-hidden="true">12.</strong> Kinematics Theory</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec12.1-continuum_motion.html"><strong aria-hidden="true">12.1.</strong> Continuum Motion</a></li><li class="chapter-item expanded "><a href="lec12.2-deformation.html"><strong aria-hidden="true">12.2.</strong> Deformation</a></li><li class="chapter-item expanded "><a href="lec12.3-summary.html"><strong aria-hidden="true">12.3.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec13-strain_energy.html"><strong aria-hidden="true">13.</strong> Strain Energy</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec13.1-rigid_null_rot_inv.html"><strong aria-hidden="true">13.1.</strong> Rigid Null Space and Rotation Invariance</a></li><li class="chapter-item expanded "><a href="lec13.2-polar_svd.html"><strong aria-hidden="true">13.2.</strong> Polar Singular Value Decomposition</a></li><li class="chapter-item expanded "><a href="lec13.3-simp_model_inversion.html"><strong aria-hidden="true">13.3.</strong> Simplified Models and Invertibility</a></li><li class="chapter-item expanded "><a href="lec13.4-summary.html"><strong aria-hidden="true">13.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec14-stress_and_derivatives.html"><strong aria-hidden="true">14.</strong> Stress and Its Derivatives</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec14.1-stress.html"><strong aria-hidden="true">14.1.</strong> Stress</a></li><li class="chapter-item expanded "><a href="lec14.2-compute_P.html"><strong aria-hidden="true">14.2.</strong> Computing Stress</a></li><li class="chapter-item expanded "><a href="lec14.3-compute_stress_deriv.html"><strong aria-hidden="true">14.3.</strong> Computing Stress Derivatives</a></li><li class="chapter-item expanded "><a href="lec14.4-summary.html"><strong aria-hidden="true">14.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec15-inv_free_elasticity.html"><strong aria-hidden="true">15.</strong> Case Study: Inversion-free Elasticity*</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec15.1-linear_tri_elem.html"><strong aria-hidden="true">15.1.</strong> Linear Triangle Elements</a></li><li class="chapter-item expanded "><a href="lec15.2-energy_grad_hess.html" class="active"><strong aria-hidden="true">15.2.</strong> Computing Energy, Gradient, and Hessian</a></li><li class="chapter-item expanded "><a href="lec15.3-filter_line_search.html"><strong aria-hidden="true">15.3.</strong> Filter Line Search for Non-Inversion</a></li><li class="chapter-item expanded "><a href="lec15.4-summary.html"><strong aria-hidden="true">15.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><li class="part-title">Governing Equations</li><li class="chapter-item expanded "><a href="lec16-strong_and_weak_forms.html"><strong aria-hidden="true">16.</strong> Strong and Weak Forms</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec16.1-mass_conserv.html"><strong aria-hidden="true">16.1.</strong> Conservation of Mass</a></li><li class="chapter-item expanded "><a href="lec16.2-momentum_conserv.html"><strong aria-hidden="true">16.2.</strong> Conservation of Momentum</a></li><li class="chapter-item expanded "><a href="lec16.3-weak_form.html"><strong aria-hidden="true">16.3.</strong> Weak Form</a></li><li class="chapter-item expanded "><a href="lec16.4-summary.html"><strong aria-hidden="true">16.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec17-disc_weak_form.html"><strong aria-hidden="true">17.</strong> Discretization of Weak Forms</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec17.1-discrete_space.html"><strong aria-hidden="true">17.1.</strong> Discrete Space</a></li><li class="chapter-item expanded "><a href="lec17.2-discrete_time.html"><strong aria-hidden="true">17.2.</strong> Discrete Time</a></li><li class="chapter-item expanded "><a href="lec17.3-summary.html"><strong aria-hidden="true">17.3.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec18-BC_and_fric.html"><strong aria-hidden="true">18.</strong> Boundary Conditions and Frictional Contact</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec18.1-incorporate_BC.html"><strong aria-hidden="true">18.1.</strong> Incorporating Boundary Conditions</a></li><li class="chapter-item expanded "><a href="lec18.2-normal_contact.html"><strong aria-hidden="true">18.2.</strong> Normal Contact for Nonpenetration</a></li><li class="chapter-item expanded "><a href="lec18.3-barrier_potential.html"><strong aria-hidden="true">18.3.</strong> Barrier Potential</a></li><li class="chapter-item expanded "><a href="lec18.4-friction_force.html"><strong aria-hidden="true">18.4.</strong> Friction Force</a></li><li class="chapter-item expanded "><a href="lec18.5-summary.html"><strong aria-hidden="true">18.5.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><li class="part-title">Finite Element Method</li><li class="chapter-item expanded "><a href="lec19-linear_FEM.html"><strong aria-hidden="true">19.</strong> Linear Finite Elements</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec19.1-linear_disp_field.html"><strong aria-hidden="true">19.1.</strong> Piecewise Linear Displacement Field</a></li><li class="chapter-item expanded "><a href="lec19.2-mass_matrix.html"><strong aria-hidden="true">19.2.</strong> Mass Matrix and Lumping</a></li><li class="chapter-item expanded "><a href="lec19.3-elasticity_term.html"><strong aria-hidden="true">19.3.</strong> Elasticity Term</a></li><li class="chapter-item expanded "><a href="lec19.4-summary.html"><strong aria-hidden="true">19.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec20-pw_linear_boundary.html"><strong aria-hidden="true">20.</strong> Piecewise Linear Boundaries</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec20.1-boundary_conditions.html"><strong aria-hidden="true">20.1.</strong> Boundary Conditions</a></li><li class="chapter-item expanded "><a href="lec20.2-obstacle_contact.html"><strong aria-hidden="true">20.2.</strong> Solid-Obstacle Contact</a></li><li class="chapter-item expanded "><a href="lec20.3-self_contact.html"><strong aria-hidden="true">20.3.</strong> Self-Contact</a></li><li class="chapter-item expanded "><a href="lec20.4-summary.html"><strong aria-hidden="true">20.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec21-2d_self_contact.html"><strong aria-hidden="true">21.</strong> Case Study: 2D Self-Contact*</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec21.1-scene_setup.html"><strong aria-hidden="true">21.1.</strong> Scene Setup and Boundary Element Collection</a></li><li class="chapter-item expanded "><a href="lec21.2-point_edge_dist.html"><strong aria-hidden="true">21.2.</strong> Point-Edge Distance</a></li><li class="chapter-item expanded "><a href="lec21.3-barrier_and_derivatives.html"><strong aria-hidden="true">21.3.</strong> Barrier Energy and Its Derivatives</a></li><li class="chapter-item expanded "><a href="lec21.4-ccd.html"><strong aria-hidden="true">21.4.</strong> Continuous Collision Detection</a></li><li class="chapter-item expanded "><a href="lec21.5-summary.html"><strong aria-hidden="true">21.5.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec22-2d_self_fric.html"><strong aria-hidden="true">22.</strong> 2D Frictional Self-Contact*</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec22.1-disc_and_approx.html"><strong aria-hidden="true">22.1.</strong> Discretization and Approximation</a></li><li class="chapter-item expanded "><a href="lec22.2-precompute.html"><strong aria-hidden="true">22.2.</strong> Precomputing Normal and Tangent Information</a></li><li class="chapter-item expanded "><a href="lec22.3-fric_and_derivatives.html"><strong aria-hidden="true">22.3.</strong> Friction Energy and Its Derivatives</a></li><li class="chapter-item expanded "><a href="lec22.4-summary.html"><strong aria-hidden="true">22.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec23-3d_elastodynamics.html"><strong aria-hidden="true">23.</strong> 3D Elastodynamics</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec23.1-kinematics.html"><strong aria-hidden="true">23.1.</strong> Kinematics</a></li><li class="chapter-item expanded "><a href="lec23.2-mass_matrix.html"><strong aria-hidden="true">23.2.</strong> Mass Matrix</a></li><li class="chapter-item expanded "><a href="lec23.3-elasticity.html"><strong aria-hidden="true">23.3.</strong> Elasticity</a></li><li class="chapter-item expanded "><a href="lec23.4-summary.html"><strong aria-hidden="true">23.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec24-3d_fric_self_contact.html"><strong aria-hidden="true">24.</strong> 3D Frictional Self-Contact</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec24.1-barrier_and_dist.html"><strong aria-hidden="true">24.1.</strong> Barrier and Distances</a></li><li class="chapter-item expanded "><a href="lec24.2-collision_detection.html"><strong aria-hidden="true">24.2.</strong> Collision Detection</a></li><li class="chapter-item expanded "><a href="lec24.3-friction.html"><strong aria-hidden="true">24.3.</strong> Friction</a></li><li class="chapter-item expanded "><a href="lec24.4-summary.html"><strong aria-hidden="true">24.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="bibliography.html">Bibliography</a></li></ol>
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<h2 id="computing-energy-gradient-and-hessian"><a class="header" href="#computing-energy-gradient-and-hessian">Computing Energy, Gradient, and Hessian</a></h2>
<p>We first follow sections <a href="lec13-strain_energy.html">Strain Energy</a> and <a href="lec14-stress_and_derivatives.html">Stress and Its Derivatives</a> to implement computing <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Ψ</span><span class="mopen">(</span><span class="mord mathbf">F</span><span class="mclose">)</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2251em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathbf mtight">F</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight">Ψ</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>, and SPD-projected <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2251em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathbf mtight">F</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathbf mtight">P</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>:</p>
<p><a name="imp:lec15:NeoHookeanEnergy_wrt_F"></a>
<strong>Implementation 15.2.1 (Energy derivatives w.r.t. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">F</span></span></span></span>, NeoHookeanEnergy.py).</strong></p>
<pre><code class="language-python">import utils
import numpy as np
import math
def polar_svd(F):
[U, s, VT] = np.linalg.svd(F)
if np.linalg.det(U) < 0:
U[:, 1] = -U[:, 1]
s[1] = -s[1]
if np.linalg.det(VT) < 0:
VT[1, :] = -VT[1, :]
s[1] = -s[1]
return [U, s, VT]
def dPsi_div_dsigma(s, mu, lam):
ln_sigma_prod = math.log(s[0] * s[1])
inv0 = 1.0 / s[0]
dPsi_dsigma_0 = mu * (s[0] - inv0) + lam * inv0 * ln_sigma_prod
inv1 = 1.0 / s[1]
dPsi_dsigma_1 = mu * (s[1] - inv1) + lam * inv1 * ln_sigma_prod
return [dPsi_dsigma_0, dPsi_dsigma_1]
def d2Psi_div_dsigma2(s, mu, lam):
ln_sigma_prod = math.log(s[0] * s[1])
inv2_0 = 1 / (s[0] * s[0])
d2Psi_dsigma2_00 = mu * (1 + inv2_0) - lam * inv2_0 * (ln_sigma_prod - 1)
inv2_1 = 1 / (s[1] * s[1])
d2Psi_dsigma2_11 = mu * (1 + inv2_1) - lam * inv2_1 * (ln_sigma_prod - 1)
d2Psi_dsigma2_01 = lam / (s[0] * s[1])
return [[d2Psi_dsigma2_00, d2Psi_dsigma2_01], [d2Psi_dsigma2_01, d2Psi_dsigma2_11]]
def B_left_coef(s, mu, lam):
sigma_prod = s[0] * s[1]
return (mu + (mu - lam * math.log(sigma_prod)) / sigma_prod) / 2
def Psi(F, mu, lam):
J = np.linalg.det(F)
lnJ = math.log(J)
return mu / 2 * (np.trace(np.transpose(F).dot(F)) - 2) - mu * lnJ + lam / 2 * lnJ * lnJ
def dPsi_div_dF(F, mu, lam):
FinvT = np.transpose(np.linalg.inv(F))
return mu * (F - FinvT) + lam * math.log(np.linalg.det(F)) * FinvT
def d2Psi_div_dF2(F, mu, lam):
[U, sigma, VT] = polar_svd(F)
Psi_sigma_sigma = utils.make_PSD(d2Psi_div_dsigma2(sigma, mu, lam))
B_left = B_left_coef(sigma, mu, lam)
Psi_sigma = dPsi_div_dsigma(sigma, mu, lam)
B_right = (Psi_sigma[0] + Psi_sigma[1]) / (2 * max(sigma[0] + sigma[1], 1e-6))
B = utils.make_PSD([[B_left + B_right, B_left - B_right], [B_left - B_right, B_left + B_right]])
M = np.array([[0, 0, 0, 0]] * 4)
M[0, 0] = Psi_sigma_sigma[0, 0]
M[0, 3] = Psi_sigma_sigma[0, 1]
M[1, 1] = B[0, 0]
M[1, 2] = B[0, 1]
M[2, 1] = B[1, 0]
M[2, 2] = B[1, 1]
M[3, 0] = Psi_sigma_sigma[1, 0]
M[3, 3] = Psi_sigma_sigma[1, 1]
dP_div_dF = np.array([[0, 0, 0, 0]] * 4)
for j in range(0, 2):
for i in range(0, 2):
ij = j * 2 + i
for s in range(0, 2):
for r in range(0, 2):
rs = s * 2 + r
dP_div_dF[ij, rs] = M[0, 0] * U[i, 0] * VT[0, j] * U[r, 0] * VT[0, s] \
+ M[0, 3] * U[i, 0] * VT[0, j] * U[r, 1] * VT[1, s] \
+ M[1, 1] * U[i, 1] * VT[0, j] * U[r, 1] * VT[0, s] \
+ M[1, 2] * U[i, 1] * VT[0, j] * U[r, 0] * VT[1, s] \
+ M[2, 1] * U[i, 0] * VT[1, j] * U[r, 1] * VT[0, s] \
+ M[2, 2] * U[i, 0] * VT[1, j] * U[r, 0] * VT[1, s] \
+ M[3, 0] * U[i, 1] * VT[1, j] * U[r, 0] * VT[0, s] \
+ M[3, 3] * U[i, 1] * VT[1, j] * U[r, 1] * VT[1, s]
return dP_div_dF
</code></pre>
<p>Next, we implement computing <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbf">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>, and the tensor products with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2251em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathbf mtight">F</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> for chain rule based computation of elasticity energy gradient and Hessian:</p>
<p><a name="imp:lec15:NeoHookeanEnergy_F_x"></a>
<strong>Implementation 15.2.2 (Energy derivatives w.r.t. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>, NeoHookeanEnergy.py).</strong></p>
<pre><code class="language-python">def deformation_grad(x, elemVInd, IB):
F = [x[elemVInd[1]] - x[elemVInd[0]], x[elemVInd[2]] - x[elemVInd[0]]]
return np.transpose(F).dot(IB)
def dPsi_div_dx(P, IB): # applying chain-rule, dPsi_div_dx = dPsi_div_dF * dF_div_dx
dPsi_dx_2 = P[0, 0] * IB[0, 0] + P[0, 1] * IB[0, 1]
dPsi_dx_3 = P[1, 0] * IB[0, 0] + P[1, 1] * IB[0, 1]
dPsi_dx_4 = P[0, 0] * IB[1, 0] + P[0, 1] * IB[1, 1]
dPsi_dx_5 = P[1, 0] * IB[1, 0] + P[1, 1] * IB[1, 1]
return [np.array([-dPsi_dx_2 - dPsi_dx_4, -dPsi_dx_3 - dPsi_dx_5]), np.array([dPsi_dx_2, dPsi_dx_3]), np.array([dPsi_dx_4, dPsi_dx_5])]
def d2Psi_div_dx2(dP_div_dF, IB): # applying chain-rule, d2Psi_div_dx2 = dF_div_dx^T * d2Psi_div_dF2 * dF_div_dx (note that d2F_div_dx2 = 0)
intermediate = np.array([[0.0, 0.0, 0.0, 0.0]] * 6)
for colI in range(0, 4):
_000 = dP_div_dF[0, colI] * IB[0, 0]
_010 = dP_div_dF[0, colI] * IB[1, 0]
_101 = dP_div_dF[2, colI] * IB[0, 1]
_111 = dP_div_dF[2, colI] * IB[1, 1]
_200 = dP_div_dF[1, colI] * IB[0, 0]
_210 = dP_div_dF[1, colI] * IB[1, 0]
_301 = dP_div_dF[3, colI] * IB[0, 1]
_311 = dP_div_dF[3, colI] * IB[1, 1]
intermediate[2, colI] = _000 + _101
intermediate[3, colI] = _200 + _301
intermediate[4, colI] = _010 + _111
intermediate[5, colI] = _210 + _311
intermediate[0, colI] = -intermediate[2, colI] - intermediate[4, colI]
intermediate[1, colI] = -intermediate[3, colI] - intermediate[5, colI]
result = np.array([[0.0, 0.0, 0.0, 0.0, 0.0, 0.0]] * 6)
for colI in range(0, 6):
_000 = intermediate[colI, 0] * IB[0, 0]
_010 = intermediate[colI, 0] * IB[1, 0]
_101 = intermediate[colI, 2] * IB[0, 1]
_111 = intermediate[colI, 2] * IB[1, 1]
_200 = intermediate[colI, 1] * IB[0, 0]
_210 = intermediate[colI, 1] * IB[1, 0]
_301 = intermediate[colI, 3] * IB[0, 1]
_311 = intermediate[colI, 3] * IB[1, 1]
result[2, colI] = _000 + _101
result[3, colI] = _200 + _301
result[4, colI] = _010 + _111
result[5, colI] = _210 + _311
result[0, colI] = -_000 - _101 - _010 - _111
result[1, colI] = -_200 - _301 - _210 - _311
return result
</code></pre>
<p>Finally, Neo-Hookean energy value, gradient, and Hessian on the entire mesh can be computed as follows:</p>
<p><a name="imp:lec15:NeoHookeanEnergy_wrt_x"></a>
<strong>Implementation 15.2.3 (Energy value, Gradient, and Hessian, NeoHookeanEnergy.py).</strong></p>
<pre><code class="language-python">def val(x, e, vol, IB, mu, lam):
sum = 0.0
for i in range(0, len(e)):
F = deformation_grad(x, e[i], IB[i])
sum += vol[i] * Psi(F, mu[i], lam[i])
return sum
def grad(x, e, vol, IB, mu, lam):
g = np.array([[0.0, 0.0]] * len(x))
for i in range(0, len(e)):
F = deformation_grad(x, e[i], IB[i])
P = vol[i] * dPsi_div_dF(F, mu[i], lam[i])
g_local = dPsi_div_dx(P, IB[i])
for j in range(0, 3):
g[e[i][j]] += g_local[j]
return g
def hess(x, e, vol, IB, mu, lam):
IJV = [[0] * (len(e) * 36), [0] * (len(e) * 36), np.array([0.0] * (len(e) * 36))]
for i in range(0, len(e)):
F = deformation_grad(x, e[i], IB[i])
dP_div_dF = vol[i] * d2Psi_div_dF2(F, mu[i], lam[i])
local_hess = d2Psi_div_dx2(dP_div_dF, IB[i])
for xI in range(0, 3):
for xJ in range(0, 3):
for dI in range(0, 2):
for dJ in range(0, 2):
ind = i * 36 + (xI * 3 + xJ) * 4 + dI * 2 + dJ
IJV[0][ind] = e[i][xI] * 2 + dI
IJV[1][ind] = e[i][xJ] * 2 + dJ
IJV[2][ind] = local_hess[xI * 2 + dI, xJ * 2 + dJ]
return IJV
</code></pre>
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