---
title: 'MLS-MPM: Moving Least Squares Material Point Method'
url: https://www.emergentmind.com/topics/moving-least-squares-material-point-method-mls-mpm
type: topic
---

# MLS-MPM: Moving Least Squares Material Point Method

The Moving Least Squares Material Point Method (MLS-MPM) is a hybrid Eulerian-Lagrangian computational approach for large-deformation solid mechanics. It augments the conventional Material Point Method (MPM) by employing local moving least squares (MLS) reconstruction to devise higher-order, adaptively smooth interpolation kernels for transferring physical quantities between particles and background grid. Recent developments extend MLS-MPM to fully unstructured meshes, generalizing its capabilities to arbitrary simplex tessellations in 2D and 3D, while ensuring continuous gradients and robust convergence—even across nonuniform geometries and extreme deformations [2312.10338][2108.00388].

## 1. Mathematical Formulation and Kernel Design

MLS-MPM recasts interpolation and gradient evaluation as localized least-squares problems centered at particle locations. For a particle at position $x$ within an unstructured simplicial mesh $\Omega$, the support of the MLS kernel is defined by the 1-ring vertex neighborhood $\mathcal{N}^1(x)$ (the element vertices and their direct neighbors), mitigating the "kernel degeneration" that occurs when support changes abruptly at cell boundaries.

A function $u(x)$ is reconstructed via a local MLS fit to its samples $u_v$ at vertices $v$ in $\mathcal{N}^1(x)$, using a linear polynomial basis $p(z) = [1, z^\top]^\top$ and a diagonal sample-weight matrix $D(x)$ with entries $D_{vv}(x) = d(x_v - x)$, where $d(\cdot)$ is smooth, compactly supported, and positive (e.g., B-spline profile). The coefficients $a(x)$ minimizing the weighted least squares error,
\[
\sum_{v \in \mathcal{N}^1(x)} d(x_v - x) \|u_v - a(x)^\top p(x_v-x)\|^2,
\]
yield the MLS reconstruction:
\[
a(x) = M(x)^{-1} b(x),
\]
where
\[
M(x) = \sum_v d(x_v-x) p(x_v-x) p(x_v-x)^\top,\qquad
b(x) = \sum_v d(x_v-x) p(x_v-x) u_v.
\]
The reconstructed value and gradient at $x$ then follow as
\[
[\hat{u}(x);\, \nabla\hat{u}(x)] = M(x)^{-1} \sum_{v\in \mathcal{N}^1(x)} d(x_v-x) p(x_v-x) u_v.
\]
To address discontinuous neighborhood transitions on unstructured meshes, the MLS kernel incorporates a diminishing function $\eta_v(x)$, vanishing smoothly as a vertex $v$ exits or enters the support:
\[
\eta_v(x) = \sum_{n \in \mathcal{N}^0(x)} B_n(x) A_{vn},
\]
where $B_n(x)$ are barycentric coordinates of $x$ in its cell, and $A_{vn}$ encodes vertex adjacency. The MLS sample weights become $\bar{w}_v(x) = \eta_v(x) d(x_v - x)$. The MLS shape function and its gradient are given by
\[
w_v(x) = p(x_v - x)^\top M(x)^{-1} \bar{w}_v(x),
\]
with the gradient derived by differentiating $w_v(x)$ with respect to $x$ [2312.10338].

## 2. Continuity, Cell-Crossing, and Gradient Properties

Standard ($\mathcal{C}^0$) piecewise-linear FEM or MPM basis functions on unstructured meshes yield discontinuous gradients across element faces, inducing "cell-crossing errors" manifested as spurious stress oscillations. The MLS-MPM approach, enhanced with the diminishing factor, achieves $\mathcal{C}^1$ continuity of both the shape functions and their gradients. This is mathematically justified: as a particle transitions from one element to another, added or removed node sets contribute weights that vanish smoothly, ensuring that both $w_v(x)$ and $\nabla w_v(x)$ remain continuous under mesh traversal—a critical requirement for accuracy and stability in high-deformation physics [2312.10338].

## 3. Algorithmic Integration: Explicit UMLS-MPM and APIC+MLS

MLS-MPM is integrated within the explicit MPM (FLIP) pipeline, substituting MLS-based functions for classical grid-to-particle and particle-to-grid transfer. The UMLS-MPM algorithm proceeds as follows:

- **Particle-to-Grid (P2G):** Mass, momentum, and internal/external forces are rasterized using the MLS shape functions $w_{ip} = w_i(x_p)$, with gradients $\nabla w_{ip}$ applied for computing stress divergence.
- **Grid Update:** Node velocities are advanced via explicit symplectic Euler integration, incorporating computed accelerations.
- **Grid-to-Particle (G2P):** Updated node velocities and accelerations are interpolated back to particle positions.
- **Deformation and Stress Update:** The deformation gradient $F_p$, particle volume, and stress tensor $\sigma_p$ are updated using the MLS kernel.
- **Reset:** Grid nodal quantities are cleared and time advances.

In the context of Arbitrary Updated Lagrangian MPM (A-ULMPM), the MLS kernel naturally interpolates between total-Lagrangian and Eulerian limits. Reference configuration updates are triggered based on a physics-driven criterion involving the deviation of the local Jacobian from unity. Kernel and support recalculations are only performed upon such updates, improving computational efficiency in scenarios with localized large deformations [2108.00388]. Velocity rasterization is further augmented by the APIC scheme, conserving angular and linear momentum up to machine precision and leveraging first-order velocity derivatives reconstructed by the MLS operator.

## 4. Stability, Robustness, and Convergence Properties

Stability derives from the partition of unity ($\sum_i w_{ip} = 1$), non-negativity (if $\eta_v, d \ge 0$ and $M$ is well-conditioned), and the continuity of MLS gradients, which suppresses stress spikes at cell crossings. Well-posedness requires non-degenerate sampling in the 1-ring support: the minimum singular value $\sigma_{\min}(M)$ must remain bounded away from zero. Empirical and theoretical analyses confirm that, under mesh gradations $R \le 1.5$, the mean-square error in particle displacement converges as $\mathrm{RMSE}(h) = O(h^2)$ for coarse grids, with temporal errors $O(\Delta t)$ under the CFL condition [2312.10338][2108.00388].

## 5. Numerical Validation and Benchmarks

Comprehensive benchmarks validate the performance and robustness of MLS-MPM and UMLS-MPM:

| Test Problem                        | Key Metrics / Observations                                                  | Reference          |
|--------------------------------------|----------------------------------------------------------------------------|--------------------|
| 1D Vibrating Bar                    | Max RMSE $<0.554\%$ @ $dx=0.5$ m; 2nd-order spatial convergence; $<0.1\%$ energy error | [2312.10338]       |
| 2D Colliding Elastic Disks           | $\leq 1\%$ momentum loss vs. high-res B-spline MPM; no spurious oscillations      | [2312.10338]       |
| 2D Rotated Cantilever                | Tip-displacement error (1.27–4.72%) for varying mesh rotations; maintains 2nd order | [2312.10338]       |
| 3D Slope Failure (Elastoplastic)     | Plasticity, mean stress, and run-out histories align with total-Lagrangian MPM | [2312.10338]       |
| 3D Spherical Container Expansion     | Conformal handling of detailed curved boundaries, impact, and expansion phases   | [2312.10338]       |
| 3D Multi-Material, Fluid/Solid Coupling | Large deformation, angular/lateral momentum conserved, adaptive reference update | [2108.00388]       |

Within these tests, UMLS-MPM consistently recovers high-order spatial convergence and eliminates cell-crossing artifacts, even for complex 2D and 3D unstructured meshes. Energy and momentum conservation metrics are consistently within strict tolerances; equivalent results on standard benchmarks match or exceed results from B-spline or structured-grid MLS-MPM [2312.10338][2108.00388].

## 6. Advantages, Limitations, and Computational Considerations

MLS-MPM and its variants retain several advantages over standard MPM approaches:

- **Adaptivity and Robustness:** The MLS kernel adapts to deformation and mesh quality, eliminating "numerical fracture" and restoring high-order convergence, even under extreme deformations.
- **Generalization to Unstructured Meshes:** UMLS-MPM exploits general simplex tessellation in both 2D and 3D, enabling accurate simulation of complex geometries.
- **Conservation Properties:** Integration with APIC methods enables exact conservation of both linear and angular momentum, up to floating-point precision [2108.00388].
- **Computational Profile:** Moment matrix inversion per particle is required upon reference reset; otherwise, most kernel operations are localized and efficient for modern accelerator hardware.
- **Reference Update Criterion:** Physics-based triggers for reference configuration change balance accuracy and computational efficiency, e.g., thresholds $\varepsilon\approx0.01$ for fluids, $0.5$ for soft solids.

*Limiting factors include potential ill-conditioning of the MLS moment matrix for very sparse or poorly graded meshes, addressed via regularization or robust solvers, and stringent timestep restrictions for highly stiff materials* [2108.00388]. Meshes with sufficiently small gradation ratios $R$ and well-defined 1-ring neighborhoods are prerequisites for stability and optimal convergence.

## 7. Connections, Extensions, and Related Methodologies

MLS-MPM generalizes the standard MPM by adapting shape function support, reproducing both constant and linear fields exactly. It provides a seamless bridge between total Lagrangian and fully Eulerian formulations, unified by the A-ULMPM approach, and is compatible with particle-in-cell schemes such as FLIP or APIC for transfer steps. The method's $\mathcal{C}^1$ kernel construction on unstructured meshes distinguishes it from prior efforts, and the diminishing factor in kernel support offers a general recipe for extending MLS-based approaches to arbitrary tessellations. Applications include elastoplasticity, multi-material interaction, and large-deformation fluid-solid coupling, making MLS-MPM a versatile tool for computational physics and engineering simulation [2312.10338][2108.00388].

---

**References:**  
- "Unstructured Moving Least Squares Material Point Methods: A Stable Kernel Approach With Continuous Gradient Reconstruction on General Unstructured Tessellations" [2312.10338]  
- "A-ULMPM: An Arbitrary Updated Lagrangian Material Point Method for Efficient Simulation of Solids and Fluids" [2108.00388]

Source: https://www.emergentmind.com/topics/moving-least-squares-material-point-method-mls-mpm