Generalized Moving Least Squares Research Articles

Generalized finite difference method on unknown manifolds

In this paper, we extend the Generalized Finite Difference Method (GFDM) on unknown compact submanifolds of the Euclidean domain, identified by randomly sampled data that (almost surely) lie on the interior of the manifolds. Theoretically, we formalize GFDM by exploiting a representation of smooth functions on the manifolds with Taylor's expansions of polynomials defined on the tangent bundles. We illustrate the approach by approximating the Laplace-Beltrami operator, where a stable approximation is achieved by a combination of Generalized Moving Least-Squares algorithm and novel linear programming that relaxes the diagonal-dominant constraint for the estimator to allow for a feasible solution even when higher-order polynomials are employed. We establish the theoretical convergence of GFDM in solving Poisson PDEs and numerically demonstrate the accuracy on simple smooth manifolds of low and moderate high co-dimensions as well as unknown 2D surfaces. For the Dirichlet Poisson problem where no data points on the boundaries are available, we employ GFDM with the volume-constraint approach that imposes the boundary conditions on data points close to the boundary. When the location of the boundary is unknown, we introduce a novel technique to detect points close to the boundary without needing to estimate the distance of the sampled data points to the boundary. We demonstrate the effectiveness of the volume-constraint employed by imposing the boundary conditions on the data points detected by this new technique compared to imposing the boundary conditions on all points within a certain distance from the boundary, where the latter is sensitive to the choice of truncation distance and requires the knowledge of the boundary location.

The MLS based numerical manifold method for bending analysis of thin plates on elastic foundations

Compared with the finite element method, H2-regularity in the Galerkin based approximation to the Kirchhoff thin plate model can be easily realized using either the moving least squares (MLS) or the generalized moving least squares (GMLS), which take the Lagrange form and the Hermite form, respectively. Coupling (G)MLS with the numerical manifold method (NMM) can greatly improve numerical properties of NMM in the treatment of plates of complicated shape, thereby denoted by MLS-NMM and GMLS-NMM. In the (G)MLS-NMM, the mathematical cover is composed of simply connected and partially overlapped mathematical patches that are the influence domains of (G)MLS nodes. GMLS-NMM appears to better fit to the Kirchhoff plate because it is equipped with rotation angle degrees of freedom. Through numerical tests and theoretical analysis in solving problems of thin plates on elastic foundations, however, this study shows that MLS-NMM is much more advantageous over GMLS-NMM from the aspects of both accuracy and memory usage.

Numerical solution of Allen–Cahn model on surfaces via an effective method based on generalized moving least squares (GMLS) approximation and the closest point approach

A numerical method has been designed for solving the surface Allen–Cahn model. The proposed numerical procedure is based on the generalized moving least-squares approximation and the closest point method. This approach is free from the structure of the underlying surface. It only needs a set of arbitrarily distributed mesh-free points on the surface.

GMLS-based numerical manifold method in mechanical analysis of thin plates with complicated shape or cutouts

We proposed a generalized moving least squares (GMLS) based numerical manifold method (NMM) and applied to investigate mechanical behaviors of thin plates with various shape and cutouts. The influence domains of nodes in generalized moving least squares (GMLS) interpolation are employed to construct the mathematical cover, which is specifically fitted for the fourth-order problems with rotational degrees of freedom. Moreover, the GMLS-based NMM can naturally treat plates with arbitrary complex geometry and openings without resorting to those complicated criteria and tailored operations while simplifying the physical cover generation process. In numerical study, a suitable mass lumping scheme is formulated and advised for transverse vibration analysis. Bending, vibration and buckling behaviors of thin plates with different shapes and cutouts are studied. Numerical results manifest that GMLS-based NMM can accurately model transverse mechanical behaviors.

Metrics for Intercomparison of Remapping Algorithms (MIRA) protocol applied to Earth system models

Abstract. Strongly coupled nonlinear phenomena such as those described by Earth system models (ESMs) are composed of multiple component models with independent mesh topologies and scalable numerical solvers. A common operation in ESMs is to remap or interpolate component solution fields defined on their computational mesh to another mesh with a different combinatorial structure and decomposition, e.g., from the atmosphere to the ocean, during the temporal integration of the coupled system. Several remapping schemes are currently in use or available for ESMs. However, a unified approach to compare the properties of these different schemes has not been attempted previously. We present a rigorous methodology for the evaluation and intercomparison of remapping methods through an independently implemented suite of metrics that measure the ability of a method to adhere to constraints such as grid independence, monotonicity, global conservation, and local extrema or feature preservation. A comprehensive set of numerical evaluations is conducted based on a progression of scalar fields from idealized and smooth to more general climate data with strong discontinuities and strict bounds. We examine four remapping algorithms with distinct design approaches, namely ESMF Regrid (Hill et al., 2004), TempestRemap (Ullrich and Taylor, 2015), generalized moving least squares (GMLS) (Trask and Kuberry, 2020) with post-processing filters, and WLS-ENOR (Li et al., 2020). By repeated iterative application of the high-order remapping methods to the test fields, we verify the accuracy of each scheme in terms of their observed convergence order for smooth data and determine the bounded error propagation using challenging, realistic field data on both uniform and regionally refined mesh cases. In addition to retaining high-order accuracy under idealized conditions, the methods also demonstrate robust remapping performance when dealing with non-smooth data. There is a failure to maintain monotonicity in the traditional L2-minimization approaches used in ESMF and TempestRemap, in contrast to stable recovery through nonlinear filters used in both meshless GMLS and hybrid mesh-based WLS-ENOR schemes. Local feature preservation analysis indicates that high-order methods perform better than low-order dissipative schemes for all test cases. The behavior of these remappers remains consistent when applied on regionally refined meshes, indicating mesh-invariant implementations. The MIRA intercomparison protocol proposed in this paper and the detailed comparison of the four algorithms demonstrate that the new schemes, namely GMLS and WLS-ENOR, are competitive compared to standard conservative minimization methods requiring computation of mesh intersections. The work presented in this paper provides a foundation that can be extended to include complex field definitions, realistic mesh topologies, and spectral element discretizations, thereby allowing for a more complete analysis of production-ready remapping packages.

A multigrid preconditioner for spatially adaptive high-order meshless method on fluid–solid interaction problems

We present a monolithic geometric multigrid preconditioner for solving fluid–solid interaction problems in Stokes limit. The problems are discretized by a spatially adaptive high-order meshless method, the generalized moving least squares (GMLS) with adaptive h-refinement. For solving fluid–solid interaction problems, we need to deal with a tightly coupled system consisting of the flow field and solid bodies, resulting in a linear system of equations with a block structure. In Stokes limit, solid kinematics can be dominated by the singularities governing the lubrication effects. Resolving those singularities with adaptive h-refinement can lead to an ill-conditioned linear system of equations. The key ingredients of the multigrid preconditioner include the interpolation and restriction operators and the smoothers. For constructing the interpolation and restriction operators, we utilize the geometric information of hierarchical sets of GMLS nodes generated in adaptive h-refinement. We build decoupled smoothers through physics-based splitting and then combine them via a multiplicative overlapping Schwarz approach. Through numerical examples with the inclusion of different numbers and shapes of solid bodies, we demonstrate the performance and assess the scalability of the designed preconditioner. As the total degrees of freedom and the number of solid bodies increase, the proposed monolithic geometric multigrid preconditioner can ensure convergence and good scalability when using the Krylov iterative method for solving the linear systems of equations generated from the spatially adaptive GMLS discretization. More specifically, for a fixed number of solid bodies, as the discretization resolution is incrementally refined, the number of iterations of the linear solver can be maintained at the same level, indicating nearly linear scalability of our preconditioner with respect to the total degrees of freedom. When the number of solid bodies Ns increases, the number of iterations is nearly proportional to Ns, implying the sublinear optimality with respect to the number of solid bodies.

First-passage time statistics on surfaces of general shape: Surface PDE solvers using Generalized Moving Least Squares (GMLS)

We develop numerical methods for computing statistics of stochastic processes on surfaces of general shape with drift-diffusion dynamics dXt=a(Xt)dt+b(Xt)dWt. We formulate descriptions of Brownian motion and general drift-diffusion processes on surfaces. We consider statistics of the form u(x)=Ex[∫0τg(Xt)dt]+Ex[f(Xτ)] for a domain Ω and the exit stopping time τ=inft⁡{t>0|Xt∉Ω}, where f,g are general smooth functions. For computing these statistics, we develop high-order Generalized Moving Least Squares (GMLS) solvers for associated surface PDE boundary-value problems based on Backward-Kolmogorov equations. We focus particularly on the mean First Passage Times (FPTs) given by the case f=0,g=1 where u(x)=Ex[τ]. We perform studies for a variety of shapes showing our methods converge with high-order accuracy both in capturing the geometry and the surface PDE solutions. We then perform studies showing how statistics are influenced by the surface geometry, drift dynamics, and spatially dependent diffusivities.

A divergence-free generalized moving least squares approximation with its application

An approximation based on moving least squares for vector-valued functions satisfying the divergence-free property was employed in [67] by Trask, Maxey, and Hu. They changed the traditional polynomial bases in moving least squares approximation to the vector-valued polynomial basis functions satisfying the divergence-free property for constructing a new vector-valued approximation. In this paper, we adopt [67], but we consider another approach, i.e., a direct method based on the generalized moving least squares (GMLS) approximation for vector-valued functions, which satisfy the divergence-free property. This GMLS approximation produces diffuse or uncertain derivatives for each component of the proposed vector-valued approximation satisfying the divergence-free property. Using the new approach, the linear functionals such as the derivatives act only on each row of the vector-valued polynomial basis functions, which reduces the computational cost comparing with the divergence-free moving least squares approximation. The pointwise error estimates of the presented approximation are obtained for the bounded sub-domains in Rd(d≥2) with an interior cone condition. Some numerical results are provided to confirm the obtained theoretical results. As an application, the Cahn-Hilliard-Hele-Shaw (CHHS) equation is solved numerically via a stabilized semi-implicit scheme in time that holds the mass conservative and energy dissipative properties and also the GMLS approximation in space. Besides, the developed vector-valued approximation has been used to compute the Leray-Helmholtz projection of the advective velocity.

Asymptotically Compatible Reproducing Kernel Collocation and Meshfree Integration for Nonlocal Diffusion

Reproducing kernel (RK) approximations are meshfree methods that construct shape functions from sets of scattered data. We present an asymptotically compatible (AC) RK collocation method for nonlocal diffusion models with Dirichlet boundary condition. The scheme is shown to be convergent to both nonlocal diffusion and its corresponding local limit as nonlocal interaction vanishes. The analysis is carried out on a special family of rectilinear Cartesian grids for linear RK method with designed kernel support. The key idea for the stability of the RK collocation scheme is to compare the collocation scheme with the standard Galerkin scheme which is stable. In addition, there is a large computational cost for assembling the stiffness matrix of the nonlocal problem because high order Gaussian quadrature is usually needed to evaluate the integral. We thus provide a remedy to the problem by introducing a quasi-discrete nonlocal diffusion operator for which no numerical quadrature is further needed after applying the RK collocation scheme. The quasi-discrete nonlocal diffusion operator combined with RK collocation is shown to be convergent to the correct local diffusion problem by taking the limits of nonlocal interaction and spatial resolution simultaneously. The theoretical results are then validated with numerical experiments. We additionally illustrate a connection between the proposed technique and an existing optimization based approach based on generalized moving least squares (GMLS).

Generalized moving least squares approximation for the solution of local and non-local models of cancer cell invasion of tissue under the effect of adhesion in one- and two-dimensional spaces

The main aim of this study is to solve numerically the mathematical models showing cancer cell invasion of tissue with/without considering the effect of cell-cell and cell-matrix adhesion. The mathematical models studied here are the systems of time-dependent reaction-diffusion-taxis equations in one- and two-dimensional spaces, which are formulated in the local and non-local forms. There are some difficulties in finding their solutions via numerical methods. The main difficulty is to compute the non-local term appearing in one of the studied models, which causes more CPU time during simulations. The current paper aims to overcome this problem, where a new meshless method, namely generalized moving least squares (GMLS) approximation in space and a semi-implicit backward differential formula of first-order (SBDF1) in time have been applied. Based on GMLS theory, the non-local term is approximated without any difficulties. Moreover, a simple method based on the GMLS technique is presented to implement the boundary conditions. The obtained discrete scheme for both mathematical models is a linear system of algebraic equations per time step. The biconjugate gradient stabilized (BiCGSTAB) algorithm with zero–fill incomplete lower–upper (ILU) preconditioner is used to solve the obtained linear system at each time step. At the end of this paper, some simulation results are reported to show the behavior of cancer cell invasion in the local model, and the non-local model due to reduction of cell-cell adhesion and increasing cell-matrix adhesion in one- and two-dimensional spaces, where two different types of distribution points have been considered in the square domain. The computational algorithms of the GMLS approximation and the developed numerical method for solving the non-local (local) model are included in the Appendix.

Meshfree methods on manifolds for hydrodynamic flows on curved surfaces: A Generalized Moving Least-Squares (GMLS) approach

We utilize generalized moving least squares (GMLS) to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds. We use exterior calculus to formulate incompressible hydrodynamic equations in the Stokesian regime and handle the divergence-free constraints via a generalized vector potential. This provides less coordinate-centric descriptions and enables the development of efficient numerical methods and splitting schemes for the fourth-order governing equations in terms of a system of second-order elliptic operators. Using a Hodge decomposition, we develop methods for manifolds having spherical topology. We show the methods exhibit high-order convergence rates for solving hydrodynamic flows on curved surfaces. The methods also provide general high-order approximations for the metric, curvature, and other geometric quantities of the manifold and associated exterior calculus operators. The approaches also can be utilized to develop high-order solvers for other scalar-valued and vector-valued problems on manifolds.

A conservative, consistent, and scalable meshfree mimetic method

Mimetic methods discretize divergence by restricting the Gauss theorem to mesh cells. Because point clouds lack such geometric entities, construction of a compatible meshfree divergence remains a challenge. In this work, we define an abstract Meshfree Mimetic Divergence (MMD) operator on point clouds by contraction of field and virtual face moments. This MMD satisfies a discrete divergence theorem, provides a discrete local conservation principle, and is first-order accurate. We consider two MMD instantiations. The first one assumes a background mesh and uses generalized moving least squares (GMLS) to obtain the necessary field and face moments. This MMD instance is appropriate for settings where a mesh is available but its quality is insufficient for a robust and accurate mesh-based discretization. The second MMD operator retains the GMLS field moments but defines virtual face moments using computationally efficient weighted graph-Laplacian equations. This MMD instance does not require a background grid and is appropriate for applications where mesh generation creates a computational bottleneck. It allows one to trade an expensive mesh generation problem for a scalable algebraic one, without sacrificing compatibility with the divergence operator. We demonstrate the approach by using the MMD operator to obtain a virtual finite-volume discretization of conservation laws on point clouds. Numerical results in the paper confirm the mimetic properties of the method and show that it behaves similarly to standard finite volume methods.

A spatially adaptive high-order meshless method for fluid–structure interactions

We present a scheme implementing an a posteriori refinement strategy in the context of a high-order meshless method for problems involving point singularities and fluid–solid interfaces. The generalized moving least squares (GMLS) discretization used in this work has been previously demonstrated to provide high-order compatible discretization of the Stokes and Darcy problems, offering a high-fidelity simulation tool for problems with moving boundaries. The meshless nature of the discretization is particularly attractive for adaptive h-refinement, especially when resolving the near-field aspects of variables and point singularities governing lubrication effects in fluid–structure interactions. We demonstrate that the resulting spatially adaptive GMLS method is able to achieve optimal convergence in the presence of singularities for both the div-grad and Stokes problems. Further, we present a series of simulations for flows of colloid suspensions, in which the refinement strategy efficiently achieved highly accurate solutions, particularly for colloids with complex geometries.

Compatible meshfree discretization of surface PDEs

Meshfree discretization of surface partial differential equations is appealing, due to their ability to naturally adapt to deforming motion of the underlying manifold. In this work, we consider an existing scheme proposed by Liang et al. reinterpreted in the context of generalized moving least squares (GMLS), showing that existing numerical analysis from the GMLS literature applies to their scheme. With this interpretation, their approach may then be unified with recent work developing compatible meshfree discretizations for the div-grad problem in $$\mathbb {R}^d$$. Informally, this is analogous to an extension of collocated finite differences to staggered finite difference methods, but in the manifold setting and with unstructured nodal data. In this way, we obtain a compatible meshfree discretization of elliptic problems on manifolds which is naturally stable for problems with material interfaces, without the need to introduce numerical dissipation or local enrichment near the interface. We provide convergence studies illustrating the high-order convergence and stability of the approach for manufactured solutions and for an adaptation of the classical five-strip benchmark to a cylindrical manifold.

Mitigation of the self-force effect in unstructured PIC codes using generalized moving least squares

Electrostatic particle-in-cell (PIC) methods use point clouds to represent a collection of charged particles and a mesh to compute the electric field generated by these particles. Such methods involve a data transfer of charge from the point cloud to the mesh and another transfer of the electric field from the mesh to the point cloud. The approximation errors introduced during these data transfers manifest as an artificial self-force acting on particles, causing a loss of momentum conservation. There exist PIC schemes on uniform Cartesian grids that provably avoid this effect. However, these schemes rely on a translation invariance property of a discrete Green’s function and cannot be extended to arbitrary unstructured grids. We present a new meshless data transfer approach that uses generalized moving least squares (GMLS) to reconstruct the electric field at particles from local mesh data on non-uniform meshes. The approach enriches the GMLS approximation space with singular functions to recover the singularity introduced at particles. This reduces the strength of the self-force by several orders of magnitude. We present results reconstructing both from linear finite elements and from Raviart–Thomas elements, demonstrating that the approach is generally applicable to PIC codes having degrees of freedom of arbitrary type.

Approximation of continuous surface differential operators with the generalized moving least-squares (GMLS) method for solving reaction–diffusion equation

In this paper, a meshless approximation based on generalized moving least squares is applied to solve the reaction–diffusion equations on the sphere and red-blood cell surfaces. The proposed method is based on the projected gradient of the shape functions, and it approximates the Laplace operator defined on the surfaces that is called Laplace–Beltrami. This technique only requires nodes at locations on the surface and the corresponding normal vectors to the surface. To discretize the time variable, an explicit time technique based on the fourth-order Runge–Kutta is used. Some numerical results on Turing and Fitzhugh–Nagumo partial differential equations are given for showing patterns which are appeared in biological phenomena.

Application of GMLS-based numerical manifold method in buckling analysis of thin plates

This paper applies the GMLS-based numerical manifold method (NMM) in analyzing the buckling behaviors of thin plates. The numerical manifold method (NMM), deploying two cover systems, namely the mathematical cover (MC) and physical cover (PC), has been successfully employed in the continuum and discontinuum deformation analysis. In this study, the MC system is constructed with influence domains in generalized moving least squares (GMLS) interpolation rather than the commonly-used finite element meshes, which is especially fitted for the fourth-order problems with rotational degrees of freedom and simplifies the possess of generating physical cover system. Moreover, the GMLS-based NMM can be readily utilized to treat plates with arbitrary complex geometry and openings without resorted to those complicated criteria and tailored operations. A series of numerical experiments concerning buckling analysis for thin plates with various shapes and cutouts have been performed, thus validating the effectiveness and accuracy of the proposed GMLS-based numerical manifold method.

Direct local boundary integral equation method for numerical solution of extended Fisher–Kolmogorov equation

In this paper, the local boundary integral equation (LBIE) method based on generalized moving least squares (GMLS) is proposed for solving extended Fisher---Kolmogorov (EFK) equation. Since the local weak forms are directly approximated from nodal data using a GMLS approximation, this method is called direct local boundary integral equation (DLBIE) method. In addition to DLBIE method, three other meshless local boundary integral equation methods such as LBIE method based on moving least squares, LBIE method based on radial point interpolation, and LBIE method based on moving Kriging are applied to find the numerical solution of EFK equation. A comparison between these methods is done from the perspective of accuracy and computational efficiency. The computational efficiency is the most significant advantage of the DLBIE method in comparison with the other local boundary integral equation methods. DLBIE shifts the numerical integrations over low-degree polynomials instead of over complicated shape functions and this reduces the computational costs, significantly. The main purpose of this paper is to show that the local boundary integral equation methods can be used for solving the fourth-order non-linear partial differential equations especially EFK equation. The numerical results confirm the good efficiency of the proposed methods for solving our model.

Comparison between two meshless methods based on collocation technique for the numerical solution of four-species tumor growth model

As is said in [27], the tumor-growth model is the incorporation of nutrient within the mixture as opposed to being modeled with an auxiliary reaction-diffusion equation. The formulation involves systems of highly nonlinear partial differential equations of surface effects through diffuse-interface models [27]. Simulations of this practical model using numerical methods can be applied for evaluating it. The present paper investigates the solution of the tumor growth model with meshless techniques. Meshless methods are applied based on the collocation technique which employ multiquadrics (MQ) radial basis function (RBFs) and generalized moving least squares (GMLS) procedures. The main advantages of these choices come back to the natural behavior of meshless approaches. As well as, a method based on meshless approach can be applied easily for finding the solution of partial differential equations in high-dimension using any distributions of points on regular and irregular domains. The present paper involves a time-dependent system of partial differential equations that describes four-species tumor growth model. To overcome the time variable, two procedures will be used. One of them is a semi-implicit finite difference method based on Crank–Nicolson scheme and another one is based on explicit Runge–Kutta time integration. The first case gives a linear system of algebraic equations which will be solved at each time-step. The second case will be efficient but conditionally stable. The obtained numerical results are reported to confirm the ability of these techniques for solving the two and three-dimensional tumor-growth equations.

Two numerical meshless techniques based on radial basis functions (RBFs) and the method of generalized moving least squares (GMLS) for simulation of coupled Klein–Gordon–Schrödinger (KGS) equations

In the present study, three numerical meshless methods are being considered to solve coupled Klein–Gordon–Schrödinger equations in one, two and three dimensions. First, the time derivative of the mentioned equation will be approximated using an implicit method based on Crank–Nicolson scheme then Kansa’s approach, RBFs-Pseudo-spectral (PS) method and generalized moving least squares (GMLS) method will be used to approximate the spatial derivatives. The proposed methods do not require any background mesh or cell structures, so they are based on a meshless approach. Applying three techniques reduces the solution of the one, two and three dimensional partial differential equations to the solution of linear system of algebraic equations. As is well-known, the use of Kansa’s approach makes the coefficients matrix in the above linear system of algebraic equations to be ill-conditioned and we applied LU decomposition technique. But when we employ PS method (Fasshauer, 2007), the matrix of coefficients in the obtained linear system of algebraic equations is well-conditioned. Also the GMLS technique yields a well-conditioned linear system, because a shifted and scaled polynomial basis will be used. At the end of this paper, we provide some examples on one, two and three-dimensions for obtaining numerical simulations. Also the obtained numerical results show the applicability of the proposed three methods to find the numerical solution of the KGS equations.