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// Populate the sidebar
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this.innerHTML = '<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="simulation-with-optimization/discrete-space-and-time/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="simulation-with-optimization/discrete-space-and-time/solid_rep.html"><strong aria-hidden="true">1.1.</strong> Representations of a Solid Geometry</a></li><li class="chapter-item expanded "><a href="simulation-with-optimization/discrete-space-and-time/newton_2nd_law.html"><strong aria-hidden="true">1.2.</strong> Newton's Second Law</a></li><li class="chapter-item expanded "><a href="simulation-with-optimization/discrete-space-and-time/time_integration.html"><strong aria-hidden="true">1.3.</strong> Time Integration</a></li><li class="chapter-item expanded "><a href="simulation-with-optimization/discrete-space-and-time/explicit_time_integration.html"><strong aria-hidden="true">1.4.</strong> Explicit Time Integration</a></li><li class="chapter-item expanded "><a href="simulation-with-optimization/discrete-space-and-time/implicit_time_integration.html"><strong aria-hidden="true">1.5.</strong> Implicit Time integration</a></li><li class="chapter-item expanded "><a href="simulation-with-optimization/discrete-space-and-time/summary.html"><strong aria-hidden="true">1.6.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="simulation-with-optimization/optimization-framework/opt_framework.html"><strong aria-hidden="true">2.</strong> Optimization Framework</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="simulation-with-optimization/optimization-framework/opt_time_integration.html"><strong aria-hidden="true">2.1.</strong> Optimization Time Integrator</a></li><li class="chapter-item expanded "><a href="simulation-with-optimization/optimization-framework/dirichlet_BC.html"><strong aria-hidden="true">2.2.</strong> Dirichlet Boundary Conditions</a></li><li class="chapter-item expanded "><a href="simulation-with-optimization/optimization-framework/contact.html"><strong aria-hidden="true">2.3.</strong> Contact</a></li><li class="chapter-item expanded "><a href="simulation-with-optimization/optimization-framework/friction.html"><strong aria-hidden="true">2.4.</strong> Friction</a></li><li class="chapter-item expanded "><a href="simulation-with-optimization/optimization-framework/summary.html"><strong aria-hidden="true">2.5.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="simulation-with-optimization/projected-newton/projected_Newton.html"><strong aria-hidden="true">3.</strong> Projected Newton</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="simulation-with-optimization/projected-newton/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="simulation-with-optimization/projected-newton/line_search.html"><strong aria-hidden="true">3.2.</strong> Line Search</a></li><li class="chapter-item expanded "><a href="simulation-with-optimization/projected-newton/grad_based_opt.html"><strong aria-hidden="true">3.3.</strong> Gradient-Based Optimization</a></li><li class="chapter-item expanded "><a href="simulation-with-optimization/projected-newton/summary.html"><strong aria-hidden="true">3.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="simulation-with-optimization/case-study-2d-mass-spring/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="simulation-with-optimization/case-study-2d-mass-spring/discretizations.html"><strong aria-hidden="true">4.1.</strong> Spatial and Temporal Discretizations</a></li><li class="chapter-item expanded "><a href="simulation-with-optimization/case-study-2d-mass-spring/inertia.html"><strong aria-hidden="true">4.2.</strong> Inertia Term</a></li><li class="chapter-item expanded "><a href="simulation-with-optimization/case-study-2d-mass-spring/mass_spring_energy.html"><strong aria-hidden="true">4.3.</strong> Mass-Spring Potential Energy</a></li><li class="chapter-item expanded "><a href="simulation-with-optimization/case-study-2d-mass-spring/opt_time_integrator.html"><strong aria-hidden="true">4.4.</strong> Optimization Time Integrator</a></li><li class="chapter-item expanded "><a href="simulation-with-optimization/case-study-2d-mass-spring/sim_with_vis.html"><strong aria-hidden="true">4.5.</strong> Simulation with Visualization</a></li><li class="chapter-item expanded "><a href="simulation-with-optimization/case-study-2d-mass-spring/gpu_accel.html"><strong aria-hidden="true">4.6.</strong> GPU-Accelerated Simulation</a></li><li class="chapter-item expanded "><a href="simulation-with-optimization/case-study-2d-mass-spring/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="boundary-treatments/dirichlet-boundary-conditions/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="boundary-treatments/dirichlet-boundary-conditions/equality_constraints.html"><strong aria-hidden="true">5.1.</strong> Equality Constraint Formulation</a></li><li class="chapter-item expanded "><a href="boundary-treatments/dirichlet-boundary-conditions/DOF_elimin.html"><strong aria-hidden="true">5.2.</strong> DOF Elimination Method</a></li><li class="chapter-item expanded "><a href="boundary-treatments/dirichlet-boundary-conditions/hanging_square.html"><strong aria-hidden="true">5.3.</strong> Case Study: Hanging Sqaure*</a></li><li class="chapter-item expanded "><a href="boundary-treatments/dirichlet-boundary-conditions/summary.html"><strong aria-hidden="true">5.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="boundary-treatments/slip-dirichlet-boundary-conditions/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="boundary-treatments/slip-dirichlet-boundary-conditions/axis_aligned.html"><strong aria-hidden="true">6.1.</strong> Axis-Aligned Slip DBC</a></li><li class="chapter-item expanded "><a href="boundary-treatments/slip-dirichlet-boundary-conditions/change_of_vars.html"><strong aria-hidden="true">6.2.</strong> Change of Variables</a></li><li class="chapter-item expanded "><a href="boundary-treatments/slip-dirichlet-boundary-conditions/general_slip_DBC.html"><strong aria-hidden="true">6.3.</strong> General Slip DBC</a></li><li class="chapter-item expanded "><a href="boundary-treatments/slip-dirichlet-boundary-conditions/summary.html"><strong aria-hidden="true">6.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="boundary-treatments/distance-barrier-for-nonpenetration/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="boundary-treatments/distance-barrier-for-nonpenetration/signed_dists.html"><strong aria-hidden="true">7.1.</strong> Signed Distances</a></li><li class="chapter-item expanded "><a href="boundary-treatments/distance-barrier-for-nonpenetration/dist_barrier_formulation.html"><strong aria-hidden="true">7.2.</strong> Distance Barrier</a></li><li class="chapter-item expanded "><a href="boundary-treatments/distance-barrier-for-nonpenetration/sol_accuracy.html"><strong aria-hidden="true">7.3.</strong> Solution Accuracy</a></li><li class="chapter-item expanded "><a href="boundary-treatments/distance-barrier-for-nonpenetration/summary.html"><strong aria-hidden="true">7.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="boundary-treatments/filter-line-search/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="boundary-treatments/filter-line-search/tunneling.html"><strong aria-hidden="true">8.1.</strong> Tunneling Issue</a></li><li class="chapter-item expanded "><a href="boundary-treatments/filter-line-search/nonpenetration_traj.html"><strong aria-hidden="true">8.2.</strong> Penetration-free Trajectory</a></li><li class="chapter-item expanded "><a href="boundary-treatments/filter-line-search/square_drop.html"><strong aria-hidden="true">8.3.</strong> Case Study: Square Drop*</a></li><li class="chapter-item expanded "><a href="boundary-treatments/filter-line-search/summary.html"><strong aria-hidden="true">8.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="boundary-treatments/frictional-contact/friction.html"><strong aria-hidden="true">9.</strong> Frictional Contact</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="boundary-treatments/frictional-contact/smooth_fric.html"><strong aria-hidden="true">9.1.</strong> Smooth Dynamic-Static Transition</a></li><li class="chapter-item expanded "><a href="boundary-treatments/frictional-contact/semi_imp_fric.html"><strong aria-hidden="true">9.2.</strong> Semi-Implicit Discretization</a></li><li class="chapter-item expanded "><a href="boundary-treatments/frictional-contact/fixed_point_iter.html"><strong aria-hidden="true">9.3.</strong> Fixed-Point Iteration</a></li><li class="chapter-item expanded "><a href="boundary-treatments/frictional-contact/summary.html"><strong aria-hidden="true">9.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="boundary-treatments/case-study-square-on-slope/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="boundary-treatments/case-study-square-on-slope/ground_to_slope.html"><strong aria-hidden="true">10.1.</strong> From Ground To Slope</a></li><li class="chapter-item expanded "><a href="boundary-treatments/case-study-square-on-slope/slope_fric.html"><strong aria-hidden="true">10.2.</strong> Slope Friction</a></li><li class="chapter-item expanded "><a href="boundary-treatments/case-study-square-on-slope/summary.html"><strong aria-hidden="true">10.3.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="boundary-treatments/moving-boundary-conditions/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="boundary-treatments/moving-boundary-conditions/penalty_method.html"><strong aria-hidden="true">11.1.</strong> Penalty Method</a></li><li class="chapter-item expanded "><a href="boundary-treatments/moving-boundary-conditions/compress_square.html"><strong aria-hidden="true">11.2.</strong> Case Study: Compressing Square*</a></li><li class="chapter-item expanded "><a href="boundary-treatments/moving-boundary-conditions/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="hyperelasticity/kinematics-theory/kinematics.html"><strong aria-hidden="true">12.</strong> Kinematics Theory</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="hyperelasticity/kinematics-theory/continuum_motion.html"><strong aria-hidden="true">12.1.</strong> Continuum Motion</a></li><li class="chapter-item expanded "><a href="hyperelasticity/kinematics-theory/deformation.html"><strong aria-hidden="true">12.2.</strong> Deformation</a></li><li class="chapter-item expanded "><a href="hyperelasticity/kinematics-theory/summary.html"><strong aria-hidden="true">12.3.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="hyperelasticity/strain-energy/strain_energy.html"><strong aria-hidden="true">13.</strong> Strain Energy</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="hyperelasticity/strain-energy/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="hyperelasticity/strain-energy/polar_svd.html"><strong aria-hidden="true">13.2.</strong> Polar Singular Value Decomposition</a></li><li class="chapter-item expanded "><a href="hyperelasticity/strain-energy/simp_model_inversion.html"><strong aria-hidden="true">13.3.</strong> Simplified Models and Invertibility</a></li><li class="chapter-item expanded "><a href="hyperelasticity/strain-energy/summary.html"><strong aria-hidden="true">13.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="hyperelasticity/stress-and-its-derivatives/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="hyperelasticity/stress-and-its-derivatives/stress.html"><strong aria-hidden="true">14.1.</strong> Stress</a></li><li class="chapter-item expanded "><a href="hyperelasticity/stress-and-its-derivatives/compute_P.html"><strong aria-hidden="true">14.2.</strong> Computing Stress</a></li><li class="chapter-item expanded "><a href="hyperelasticity/stress-and-its-derivatives/compute_stress_deriv.html"><strong aria-hidden="true">14.3.</strong> Computing Stress Derivatives</a></li><li class="chapter-item expanded "><a href="hyperelasticity/stress-and-its-derivatives/summary.html"><strong aria-hidden="true">14.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="hyperelasticity/case-study-inversion-free-elasticity/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="hyperelasticity/case-study-inversion-free-elasticity/linear_tri_elem.html"><strong aria-hidden="true">15.1.</strong> Linear Triangle Elements</a></li><li class="chapter-item expanded "><a href="hyperelasticity/case-study-inversion-free-elasticity/energy_grad_hess.html"><strong aria-hidden="true">15.2.</strong> Computing Energy, Gradient, and Hessian</a></li><li class="chapter-item expanded "><a href="hyperelasticity/case-study-inversion-free-elasticity/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="hyperelasticity/case-study-inversion-free-elasticity/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="governing-equations/strong-and-weak-forms/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="governing-equations/strong-and-weak-forms/mass_conserv.html"><strong aria-hidden="true">16.1.</strong> Conservation of Mass</a></li><li class="chapter-item expanded "><a href="governing-equations/strong-and-weak-forms/momentum_conserv.html"><strong aria-hidden="true">16.2.</strong> Conservation of Momentum</a></li><li class="chapter-item expanded "><a href="governing-equations/strong-and-weak-forms/weak_form.html"><strong aria-hidden="true">16.3.</strong> Weak Form</a></li><li class="chapter-item expanded "><a href="governing-equations/strong-and-weak-forms/summary.html"><strong aria-hidden="true">16.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="governing-equations/discretization-of-weak-forms/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="governing-equations/discretization-of-weak-forms/discrete_space.html"><strong aria-hidden="true">17.1.</strong> Discrete Space</a></li><li class="chapter-item expanded "><a href="governing-equations/discretization-of-weak-forms/discrete_time.html"><strong aria-hidden="true">17.2.</strong> Discrete Time</a></li><li class="chapter-item expanded "><a href="governing-equations/discretization-of-weak-forms/summary.html"><strong aria-hidden="true">17.3.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="governing-equations/boundary-conditions-and-frictional-contact/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="governing-equations/boundary-conditions-and-frictional-contact/incorporate_BC.html"><strong aria-hidden="true">18.1.</strong> Incorporating Boundary Conditions</a></li><li class="chapter-item expanded "><a href="governing-equations/boundary-conditions-and-frictional-contact/normal_contact.html"><strong aria-hidden="true">18.2.</strong> Normal Contact for Nonpenetration</a></li><li class="chapter-item expanded "><a href="governing-equations/boundary-conditions-and-frictional-contact/barrier_potential.html"><strong aria-hidden="true">18.3.</strong> Barrier Potential</a></li><li class="chapter-item expanded "><a href="governing-equations/boundary-conditions-and-frictional-contact/friction_force.html"><strong aria-hidden="true">18.4.</strong> Friction Force</a></li><li class="chapter-item expanded "><a href="governing-equations/boundary-conditions-and-frictional-contact/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="finite-element-method/linear-finite-elements/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="finite-element-method/linear-finite-elements/linear_disp_field.html"><strong aria-hidden="true">19.1.</strong> Piecewise Linear Displacement Field</a></li><li class="chapter-item expanded "><a href="finite-element-method/linear-finite-elements/mass_matrix.html"><strong aria-hidden="true">19.2.</strong> Mass Matrix and Lumping</a></li><li class="chapter-item expanded "><a href="finite-element-method/linear-finite-elements/elasticity_term.html"><strong aria-hidden="true">19.3.</strong> Elasticity Term</a></li><li class="chapter-item expanded "><a href="finite-element-method/linear-finite-elements/summary.html"><strong aria-hidden="true">19.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="finite-element-method/piecewise-linear-boundaries/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="finite-element-method/piecewise-linear-boundaries/boundary_conditions.html"><strong aria-hidden="true">20.1.</strong> Boundary Conditions</a></li><li class="chapter-item expanded "><a href="finite-element-method/piecewise-linear-boundaries/obstacle_contact.html"><strong aria-hidden="true">20.2.</strong> Solid-Obstacle Contact</a></li><li class="chapter-item expanded "><a href="finite-element-method/piecewise-linear-boundaries/self_contact.html"><strong aria-hidden="true">20.3.</strong> Self-Contact</a></li><li class="chapter-item expanded "><a href="finite-element-method/piecewise-linear-boundaries/summary.html"><strong aria-hidden="true">20.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="finite-element-method/case-study-2d-self-contact/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="finite-element-method/case-study-2d-self-contact/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="finite-element-method/case-study-2d-self-contact/point_edge_dist.html"><strong aria-hidden="true">21.2.</strong> Point-Edge Distance</a></li><li class="chapter-item expanded "><a href="finite-element-method/case-study-2d-self-contact/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="finite-element-method/case-study-2d-self-contact/ccd.html"><strong aria-hidden="true">21.4.</strong> Continuous Collision Detection</a></li><li class="chapter-item expanded "><a href="finite-element-method/case-study-2d-self-contact/summary.html"><strong aria-hidden="true">21.5.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="finite-element-method/2d-frictional-self-contact/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="finite-element-method/2d-frictional-self-contact/disc_and_approx.html"><strong aria-hidden="true">22.1.</strong> Discretization and Approximation</a></li><li class="chapter-item expanded "><a href="finite-element-method/2d-frictional-self-contact/precompute.html"><strong aria-hidden="true">22.2.</strong> Precomputing Normal and Tangent Information</a></li><li class="chapter-item expanded "><a href="finite-element-method/2d-frictional-self-contact/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="finite-element-method/2d-frictional-self-contact/summary.html"><strong aria-hidden="true">22.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="finite-element-method/3d-elastodynamics/3d_elastodynamics.html"><strong aria-hidden="true">23.</strong> 3D Elastodynamics</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="finite-element-method/3d-elastodynamics/kinematics.html"><strong aria-hidden="true">23.1.</strong> Kinematics</a></li><li class="chapter-item expanded "><a href="finite-element-method/3d-elastodynamics/mass_matrix.html"><strong aria-hidden="true">23.2.</strong> Mass Matrix</a></li><li class="chapter-item expanded "><a href="finite-element-method/3d-elastodynamics/elasticity.html"><strong aria-hidden="true">23.3.</strong> Elasticity</a></li><li class="chapter-item expanded "><a href="finite-element-method/3d-elastodynamics/summary.html"><strong aria-hidden="true">23.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="finite-element-method/3d-frictional-self-contact/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="finite-element-method/3d-frictional-self-contact/barrier_and_dist.html"><strong aria-hidden="true">24.1.</strong> Barrier and Distances</a></li><li class="chapter-item expanded "><a href="finite-element-method/3d-frictional-self-contact/collision_detection.html"><strong aria-hidden="true">24.2.</strong> Collision Detection</a></li><li class="chapter-item expanded "><a href="finite-element-method/3d-frictional-self-contact/friction.html"><strong aria-hidden="true">24.3.</strong> Friction</a></li><li class="chapter-item expanded "><a href="finite-element-method/3d-frictional-self-contact/summary.html"><strong aria-hidden="true">24.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><li class="part-title">Spatial Reductions</li><li class="chapter-item expanded "><a href="spatial-reductions/rigid-body-simulation/rigid_body_sim.html"><strong aria-hidden="true">25.</strong> Rigid Body Simulation*</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="spatial-reductions/rigid-body-simulation/rigid_body_dynamics.html"><strong aria-hidden="true">25.1.</strong> Rigid Body Dynamics</a></li><li class="chapter-item expanded "><a href="spatial-reductions/rigid-body-simulation/subspace_sim.html"><strong aria-hidden="true">25.2.</strong> Subspace Simulation</a></li><li class="chapter-item expanded "><a href="spatial-reductions/rigid-body-simulation/affine_body_dynamics.html"><strong aria-hidden="true">25.3.</strong> Affine Body Dynamics</a></li><li class="chapter-item expanded "><a href="spatial-reductions/rigid-body-simulation/dropping_block.html"><strong aria-hidden="true">25.4.</strong> Case Study: ABD Square Drop*</a></li><li class="chapter-item expanded "><a href="spatial-reductions/rigid-body-simulation/summary.html"><strong aria-hidden="true">25.5.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="spatial-reductions/modal-reductions/modal_reductions.html"><strong aria-hidden="true">26.</strong> Modal Reductions*</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="spatial-reductions/modal-reductions/linear_modal_analysis.html"><strong aria-hidden="true">26.1.</strong> Linear Modal Analysis</a></li><li class="chapter-item expanded "><a href="spatial-reductions/modal-reductions/linear_demo.html"><strong aria-hidden="true">26.2.</strong> Linear Demo*</a></li><li class="chapter-item expanded "><a href="spatial-reductions/modal-reductions/nonlinear_reductions.html"><strong aria-hidden="true">26.3.</strong> Non-linear Reductions</a></li><li class="chapter-item expanded "><a href="spatial-reductions/modal-reductions/basis_choice.html"><strong aria-hidden="true">26.4.</strong> Reduction Basis</a></li><li class="chapter-item expanded "><a href="spatial-reductions/modal-reductions/summary.html"><strong aria-hidden="true">26.5.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><li class="part-title">Material Point Method</li><li class="chapter-item expanded "><a href="material-point-method/spatial-and-temporal-discretization/mpm_disc.html"><strong aria-hidden="true">27.</strong> Spatial and Temporal Discretization</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="material-point-method/spatial-and-temporal-discretization/material_particles.html"><strong aria-hidden="true">27.1.</strong> Material Particles</a></li><li class="chapter-item expanded "><a href="material-point-method/spatial-and-temporal-discretization/interpolating_functions.html"><strong aria-hidden="true">27.2.</strong> Interpolating Functions</a></li><li class="chapter-item expanded "><a href="material-point-method/spatial-and-temporal-discretization/euler_lag_transfers.html"><strong aria-hidden="true">27.3.</strong> Particle-Grid Transfers</a></li><li class="chapter-item expanded "><a href="material-point-method/spatial-and-temporal-discretization/particle_state_update.html"><strong aria-hidden="true">27.4.</strong> Deformation Gradient and Particle State Update</a></li><li class="chapter-item expanded "><a href="material-point-method/spatial-and-temporal-discretization/summary.html"><strong aria-hidden="true">27.5.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="material-point-method/beyond-elasticity-plasticity-and-viscosity/mpm_plasticity.html"><strong aria-hidden="true">28.</strong> Beyond Elasticity: Plasticity and Viscosity</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="material-point-method/beyond-elasticity-plasticity-and-viscosity/disc_plastic_flow.html"><strong aria-hidden="true">28.1.</strong> Discretization of Plastic Flow</a></li><li class="chapter-item expanded "><a href="material-point-method/beyond-elasticity-plasticity-and-viscosity/yield_criterion.html"><strong aria-hidden="true">28.2.</strong> Yield Condition and Return Mapping</a></li><li class="chapter-item expanded "><a href="material-point-method/beyond-elasticity-plasticity-and-viscosity/summary.html"><strong aria-hidden="true">28.3.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="material-point-method/boundary-treatments/mpm_bc.html"><strong aria-hidden="true">29.</strong> Boundary Treatments</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="material-point-method/boundary-treatments/grid_bc.html"><strong aria-hidden="true">29.1.</strong> Boundary Conditions</a></li><li class="chapter-item expanded "><a href="material-point-method/boundary-treatments/contact_on_particles.html"><strong aria-hidden="true">29.2.</strong> Frictional Contact on Material Particles</a></li><li class="chapter-item expanded "><a href="material-point-method/boundary-treatments/summary.html"><strong aria-hidden="true">29.3.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="material-point-method/case-study-two-colliding-elastic-blocks-in-2d/mpm_elastic_case_study.html"><strong aria-hidden="true">30.</strong> Case Study: Two Colliding Elastic Blocks in 2D*</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="material-point-method/case-study-two-colliding-elastic-blocks-in-2d/simulation_setup.html"><strong aria-hidden="true">30.1.</strong> Simulation Setup</a></li><li class="chapter-item expanded "><a href="material-point-method/case-study-two-colliding-elastic-blocks-in-2d/pic_transfer.html"><strong aria-hidden="true">30.2.</strong> Particle-In-Cell Transfer</a></li><li class="chapter-item expanded "><a href="material-point-method/case-study-two-colliding-elastic-blocks-in-2d/summary.html"><strong aria-hidden="true">30.3.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="material-point-method/case-study-plastic-material-models/mpm_plastic_models.html"><strong aria-hidden="true">31.</strong> Case Study: Plastic Material Models*</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="material-point-method/case-study-plastic-material-models/drucker_prager.html"><strong aria-hidden="true">31.1.</strong> Drucker-Prager Elastoplasticity</a></li><li class="chapter-item expanded "><a href="material-point-method/case-study-plastic-material-models/sphere_sdf.html"><strong aria-hidden="true">31.2.</strong> SDF-based Sphere Collider</a></li><li class="chapter-item expanded "><a href="material-point-method/case-study-plastic-material-models/apic_transfer.html"><strong aria-hidden="true">31.3.</strong> Affine Particle-In-Cell Transfer</a></li><li class="chapter-item expanded "><a href="material-point-method/case-study-plastic-material-models/2d_viscoplastic_bars.html"><strong aria-hidden="true">31.4.</strong> 2D Viscoplastic Bars</a></li><li class="chapter-item expanded "><a href="material-point-method/case-study-plastic-material-models/summary.html"><strong aria-hidden="true">31.5.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><li class="part-title">Position Based Simulations</li><li class="chapter-item expanded "><a href="position-based-simulations/position-based-dynamics-pbd/position_based_dynamics.html"><strong aria-hidden="true">32.</strong> Position Based Dynamics (PBD)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="position-based-simulations/position-based-dynamics-pbd/pbd_preliminaries.html"><strong aria-hidden="true">32.1.</strong> Particle Simulation Preliminaries</a></li><li class="chapter-item expanded "><a href="position-based-simulations/position-based-dynamics-pbd/core_framework.html"><strong aria-hidden="true">32.2.</strong> Core Algorithm / Position Based Framework</a></li><li class="chapter-item expanded "><a href="position-based-simulations/position-based-dynamics-pbd/formulation.html"><strong aria-hidden="true">32.3.</strong> Constraint Formulation</a></li><li class="chapter-item expanded "><a href="position-based-simulations/position-based-dynamics-pbd/solver.html"><strong aria-hidden="true">32.4.</strong> Constraint Solver</a></li><li class="chapter-item expanded "><a href="position-based-simulations/position-based-dynamics-pbd/stiffness_damping.html"><strong aria-hidden="true">32.5.</strong> Extended PBD (XPBD): Stiffness and Damping</a></li><li class="chapter-item expanded "><a href="position-based-simulations/position-based-dynamics-pbd/summary.html"><strong aria-hidden="true">32.6.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="position-based-simulations/case-study-cloth-simulation/cloth_pbd.html"><strong aria-hidden="true">33.</strong> Case Study: Cloth Simulation*</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="position-based-simulations/case-study-cloth-simulation/stretching_constraints.html"><strong aria-hidden="true">33.1.</strong> Stretching Constraints</a></li><li class="chapter-item expanded "><a href="position-based-simulations/case-study-cloth-simulation/bending_constraints.html"><strong aria-hidden="true">33.2.</strong> Bending Constraints</a></li><li class="chapter-item expanded "><a href="position-based-simulations/case-study-cloth-simulation/cloth_framework.html"><strong aria-hidden="true">33.3.</strong> Complete Cloth Simulation</a></li></ol></li><li class="chapter-item expanded "><a href="position-based-simulations/case-study-mesh-simulation/mesh_pbd.html"><strong aria-hidden="true">34.</strong> Case Study: Mesh Simulation*</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="position-based-simulations/case-study-mesh-simulation/volume_constraints.html"><strong aria-hidden="true">34.1.</strong> Volume Constraints</a></li><li class="chapter-item expanded "><a href="position-based-simulations/case-study-mesh-simulation/edge_constraints.html"><strong aria-hidden="true">34.2.</strong> Edge Constraints</a></li><li class="chapter-item expanded "><a href="position-based-simulations/case-study-mesh-simulation/mesh_framework.html"><strong aria-hidden="true">34.3.</strong> Complete Mesh Simulation</a></li></ol></li><li class="chapter-item expanded "><a href="position-based-simulations/case-study-position-based-fluids/fluids_pbd.html"><strong aria-hidden="true">35.</strong> Case Study: Position Based Fluids*</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="position-based-simulations/case-study-position-based-fluids/density_constraints.html"><strong aria-hidden="true">35.1.</strong> Density Constraints</a></li><li class="chapter-item expanded "><a href="position-based-simulations/case-study-position-based-fluids/tensile_instability.html"><strong aria-hidden="true">35.2.</strong> Tensile Instability Correction</a></li><li class="chapter-item expanded "><a href="position-based-simulations/case-study-position-based-fluids/advanced_effects.html"><strong aria-hidden="true">35.3.</strong> Advanced Fluid Effects</a></li><li class="chapter-item expanded "><a href="position-based-simulations/case-study-position-based-fluids/pbf_framework.html"><strong aria-hidden="true">35.4.</strong> Complete PBF Simulation</a></li></ol></li><li class="chapter-item expanded "><a href="position-based-simulations/other-common-constraints/other-constraints.html"><strong aria-hidden="true">36.</strong> Other Common Constraints</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="position-based-simulations/other-common-constraints/pbd_collisions_constraints.html"><strong aria-hidden="true">36.1.</strong> Collision Constraints</a></li><li class="chapter-item expanded "><a href="position-based-simulations/other-common-constraints/pbd_continuum.html"><strong aria-hidden="true">36.2.</strong> Continuum-based Constraints</a></li><li class="chapter-item expanded "><a href="position-based-simulations/other-common-constraints/summary.html"><strong aria-hidden="true">36.3.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><li class="part-title">Special Topics</li><li class="chapter-item expanded "><a href="special-topics/linear-system/linear_sys.html"><strong aria-hidden="true">37.</strong> Linear System</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="special-topics/linear-system/direct_solver.html"><strong aria-hidden="true">37.1.</strong> Direct Solver</a></li><li class="chapter-item expanded "><a href="special-topics/linear-system/basic_iterative_methods.html"><strong aria-hidden="true">37.2.</strong> Basic Iterative Methods</a></li><li class="chapter-item expanded "><a href="special-topics/linear-system/conjugate_gradient.html"><strong aria-hidden="true">37.3.</strong> Conjugate Gradient</a></li><li class="chapter-item expanded "><a href="special-topics/linear-system/summary.html"><strong aria-hidden="true">37.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="bibliography.html">Bibliography</a></li></ol>';
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