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<title>Representations of a Solid Geometry - 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" class="active"><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"><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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<h1 class="menu-title">Physics-Based Simulation</h1>
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<h2 id="representations-of-a-solid-geometry"><a class="header" href="#representations-of-a-solid-geometry">Representations of a Solid Geometry</a></h2>
<p>In everyday life, solid objects are perceived as continuous. Yet, in the digital world of computers, where we use discrete numbers for representation, a range of interesting methods arises.</p>
<p>One method is <strong>parametrization</strong>. Consider a 3D sphere, which can be described as <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="mopen">{</span><span class="mord mathbf">x</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:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span><span class="mspace"> </span><span class="mord">∣</span><span class="mspace"> </span><span class="mord">∥</span><span class="mord mathbf">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbf">c</span><span class="mord">∥</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:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace"> </span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbf">c</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:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace"> </span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</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:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mclose">}</span></span></span></span>, centered at point \( \mathbf{c} \) with radius \( r \). This approach extends beyond spheres to include shapes like half-spaces, boxes, ellipsoids, tori, and others, characterized by their <em>interior</em> using functions such as signed distances. However, parametrization faces challenges when handling complex geometries that are frequently encountered in real-world scenarios. An emerging exception to this limitation is the use of advanced neural representations employing neural networks. These newer methods show promise in effectively representing more intricate geometrical forms.</p>
<p>An alternative is <strong>representing with sampling</strong>. This involves choosing points on and inside the object. But points alone aren't enough; we typically need to establish connectivity between them to define the object’s boundaries for applications like rendering and 3D printing. Monitoring how a cluster of points shifts over time also helps in measuring deformation.</p>
<p>In continuum mechanics, an object is seen as having a continuous density field. Digitally, this continuity must be represented discretely, usually through defining the <strong>connectivity</strong> of the solid's geometry.</p>
<blockquote>
<p><strong><a name="rem:lec1:other_solid_rep"></a>
<em>Remark 1.1.1 (Other Solid Representations).</em></strong>
There are other methods for representing solid geometries, such as voxel-based approaches. These methods divide the space into a 3D grid of small boxes, or voxels, with each voxel representing a segment of the object, similar to pixels in a 2D image. Voxel-based methods are advantageous for several reasons. Firstly, they can act as a discrete level set representation, capable of modeling complex geometries and tracking their evolution over time. Each voxel contains information about its position relative to the object's surface, offering an efficient discrete approximation of the continuous level set function. This is beneficial for algorithms involved in surface evolution, shape optimization, and collision detection. Secondly, voxel-based approaches are conducive to Constructive Solid Geometry (CSG) operations. This technique in solid modeling uses Boolean operators to combine simpler shapes into complex 3D models. The voxelized framework allows for straightforward and efficient execution of operations like union, intersection, and difference on the voxel grid. This enables the easy creation and modification of intricate shapes.</p>
</blockquote>
<blockquote>
<p><strong><a name="exp:lec1:mesh"></a>
<strong>Example 1.1.1 (Mesh).</strong></strong>
The method of creating a <strong>mesh</strong> by directly connecting points with edges or triangles is a popular technique in computational geometry. This concept is illustrated in the accompanying figure, where the left and middle images show two different meshes. Notably, even though these meshes utilize the same sampled points or nodes, they have distinct connectivities, resulting in different shapes. The rightmost mesh in the figure demonstrates a transformation from one shape to another. This mesh represents a deformation of the middle mesh, achieved by vertically compressing its upper half. <figure><center><img src="img/lec1/mesh.png" width="350"><figcaption><b><a name="fig:lec1:mesh"></a>
Figure 1.1.1.</b> Mesh</figcaption></center></figure></p>
</blockquote>
<blockquote>
<p><strong><a name="exp:lec1:particle_grid"></a>
<strong>Example 1.1.2 (Particle and Grid).</strong></strong>
By implementing a uniform grid structure in our spatial representation, we record the extent of solid matter at each node location. This allows us to use our sampled points to calculate the density of the solid at each grid node. This method is beneficial for quantifying the solid's distribution within the grid and for establishing a network of connectivity among the original sampled points. Refer to the accompanying figure for a visual demonstration of this concept. In the figure, the sampled points are depicted as green dots. The grid nodes, where we record solid densities, are shown as black circles. These nodes are connected through the grid, illustrated with blue lines. <figure><center><img src="img/lec1/MPM.png" width="250"><figcaption><b><a name="fig:lec1:particle_grid"></a>
Figure 1.1.2.</b> Particle and grid</figcaption></center></figure></p>
</blockquote>
<p>In the field of modern solid simulation, the described methods of defining connectivity are crucial. The first method, establishing connections through a mesh of edges or triangles, is foundational to Finite Element Method (FEM) simulators. The second approach, which involves using a uniform grid to compute solid density and establish connectivity, is integral to Material Point Method (MPM) simulators <a href="bibliography.html#jiang2016material">[Jiang et al. 2016]</a>. This book largely concentrates on the former method, delving into the intricacies of FEM. The mesh-based structure of FEM is particularly effective in handling complex domains by breaking them down into simpler elements. This makes FEM an essential tool in the study and simulation of deformable solids, and understanding its nuances is vital for those engaged in this area of study.</p>
<p>At first glance, the use of two representations of solid geometry in the MPM might appear redundant. Yet, this dual approach gives MPM a significant edge, especially in simulating dynamic events like solid fractures. In such cases, FEM would necessitate meticulous modification of the edges and elements that define the original connectivity to accurately depict the damage. In contrast, MPM efficiently handles these scenarios. The uniform grid naturally accommodates the separation of body parts in a fracture, as the lack of material at fracture nodes leads to an automatic disconnection of adjacent grid nodes. This attribute allows MPM to excel in managing changes in solid topology.</p>
<p>However, when it comes to simulation accuracy control, the Finite Element Method (FEM) excels. FEM operates directly on the mesh, obviating the need for constant information transfer, thus ensuring greater precision. This level of accuracy makes FEM an invaluable resource in the precise simulation of deformable solids, which is the primary emphasis of this book.</p>
<p>The technique of consolidating coordinates of each sampled point into an extended vector, denoted as \( x\in\mathbb{R}^{dn} \) (refer to the figure below), provides an effective means to describe a specific geometric configuration, given a constant connectivity. In this representation, \(d\) indicates the dimension of space (1, 2, or 3), and \(n\) represents the total number of points. Similarly, attributes like velocity, acceleration, and forces at each sample point can be amalgamated into corresponding extended vectors, symbolized as \(v\), \(a\), and \(f\) respectively. This organized approach to data presentation not only aids in comprehensively understanding the various parameters and their interrelations but also streamlines the mathematical formulation of the simulation process.</p>
<figure><center><img src="img/lec1/stacked_var.png" width="300"><figcaption><b><a name="fig:lec1:stacked_var"></a>
Figure 1.1.3.</b> Stacked position vector</center></figcaption></figure>
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