Yiming Ying

P, the key contribution lies in our implicit grid update. Within-step Momentum Residual. Instead of accepting the outcome of an explicit update, we implicitly seek an end-of-step state that is consistent with the impulse accumulated within a time step. To achieve this, we treat the volume Ωt\Omega_{t} as a whole during trial movement to predict its potential outcome. Thus, based on the strong-form momentum balance in Eq. (3), we have the weak form for grid node II: ∫Ωtρ​NI​𝐯˙​𝑑Ω=−∫Ωt∇NI:𝝈​d​Ω+∫ΩtNI​𝐛​𝑑Ω+∫∂ΩtNI​𝐭​𝑑S,\int_{\Omega_{t}}\!\!\rho N_{I}\dot{\mathbf{v}}d\Omega=-\!\!\int_{\Omega_{t}}\!\!\nabla N_{I}:\boldsymbol{\sigma}d\Omega+\int_{\Omega_{t}}\!\!N_{I}\mathbf{b}d\Omega+\int_{\partial\Omega_{t}}\!\!\!N_{I}\mathbf{t}dS, (7) where NIN_{I} is the grid shape function, 𝐛\mathbf{b} is body-force density, and 𝐭\mathbf{t} is prescribed traction. Following implicit MPM formulations Guilkey and Weiss (2003); Sulsky and Kaul (2004), we discretize the dynamics on the background grid and use a Newmark family update parameterized by (β,γ)(\beta,\gamma). Critically, we aim to predict a trial state for the next step. Therefore, we introduce grid displacement increment Δ​𝐮I\Delta\mathbf{u}_{I} over a step [tn,tn+1][t_{n},t_{n+1}] to understand how the current movement is affected by the mechanics. Particularly, considering an non-Dirichlet-constrained free node I∈ℱI\in\mathcal{F}, we induce the end-of-step kinematics based on the Newmark relations: 𝐚In+1​(Δ​𝐮I)=Δ​𝐮I−Δ​t​𝐯In−Δ​t2​(12−β)​𝐚Inβ​Δ​t2,\displaystyle\mathbf{a}_{I}^{n+1}(\Delta\mathbf{u}_{I})\!=\!\frac{\Delta\mathbf{u}_{I}-\Delta t\,\mathbf{v}_{I}^{n}-\Delta t^{2}\left(\frac{1}{2}-\beta\right)\mathbf{a}_{I}^{n}}{\beta\,\Delta t^{2}}, (8) 𝐯In+1​(Δ​𝐮I)=𝐯In+Δ​t​((1−γ)​𝐚In+γ​𝐚In+1​(Δ​𝐮I)).\displaystyle\mathbf{v}_{I}^{n+1}(\Delta\mathbf{u}_{I})\!=\!\mathbf{v}_{I}^{n}\!+\!\Delta t\Big((1\!-\!\gamma)\mathbf{a}_{I}^{n}\!+\!\gamma\,\mathbf{a}_{I}^{n+1}(\Delta\mathbf{u}_{I})\Big). (9) To implicitly analyze the end-of-step trial, we quantify the mismatch between current inertia and the sum of internal and external forces via a within-step momentum residual: 𝐑I​(Δ​𝐮)=𝐟Iext​(Δ​𝐮)+𝐟Iint​(Δ​𝐮)−mI​𝐚In+1​(Δ​𝐮I),I∈ℱ,\mathbf{R}_{I}(\Delta\mathbf{u})=\mathbf{f}^{\mathrm{ext}}_{I}(\Delta\mathbf{u})+\mathbf{f}^{\mathrm{int}}_{I}(\Delta\mathbf{u})-m_{I}\,\mathbf{a}_{I}^{n+1}(\Delta\mathbf{u}_{I}),\ I\in\mathcal{F}, (10) where mIm_{I} is the lumped grid mass for a grid node, and the internal force 𝐟Iint​(Δ​𝐮)\mathbf{f}^{\mathrm{int}}_{I}(\Delta\mathbf{u}) is accumulated from particles using the trial deformation and stress: 𝐟Iint​(Δ​𝐮)=−∑pVp0​𝐏p​(Δ​𝐮)​∇𝐗NI​(𝐱p),𝐏p=𝝉p​𝐅p−T,\mathbf{f}^{\mathrm{int}}_{I}(\Delta\mathbf{u})\!=\!-\!\sum_{p}V_{p}^{0}\,\mathbf{P}_{p}(\Delta\mathbf{u})\,\nabla_{\mathbf{X}}N_{I}(\mathbf{x}_{p}),\ \mathbf{P}_{p}\!=\!\boldsymbol{\tau}_{p}\,\mathbf{F}_{p}^{-T},\! (11) where Vp