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# Compatible meshfree discretization of surface PDEs

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## Abstract

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 Rd. 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.

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... On the other hand, several meshfree discretizations of plates and shells have been formulated [87][88][89][90][91][92][93][94]. In the broader meshfree discretization literature, a number of works have been developed for the discretization of general surface PDEs [95][96][97][98][99][100][101][102][103][104][105][106][107]. Such schemes treat the manifold extrinsically, working in ambient space and projecting to a manifold, or intrinsically, using compact reconstructions of the manifold to obtain local coordinates. ...
... Such schemes treat the manifold extrinsically, working in ambient space and projecting to a manifold, or intrinsically, using compact reconstructions of the manifold to obtain local coordinates. The latter generally apply polynomial reconstruction to obtain estimates of the metric tensor necessary for computing surface differential operators [95][96][97][108][109][110]. In contrast, by working in the reference configuration, the linear PCA reconstruction used in the current approach provides high-order treatment of surface curvature, avoiding the need for either high-order polynomial reconstruction or use of greater than C 0 continuity shape functions. ...
... We start with a meshfree, local parameterization of the shell mid-surface. We employ the approach detailed in [96] to define a local parameterization of a point cloud without resorting to using a global parametric map. Considering the undeformed configuration of a mid-surface B 2D , we approximate a tangent plane at each surface mesh point P ∈ B 2D using Principal Component Analysis (PCA) [59]. ...
Article
We present a comprehensive rotation-free Kirchhoff–Love (KL) shell formulation for peridynamics (PD) that is capable of modeling large elasto-plastic deformations and fracture in thin-walled structures. To remove the need for a predefined global parametric domain, Principal Component Analysis is employed in a meshfree setting to develop a local parameterization of the shell midsurface. The KL shell kinematics is utilized to develop a correspondence-based PD formulation. A bond-stabilization technique is employed to naturally achieve stability of the discrete solution. Only the mid-surface velocity degrees of freedom are used in the governing thin-shell equations. 3D rate-form material models are employed to enable simulating a wide range of material behavior. A bond-associative damage correspondence modeling approach is adopted to use classical failure criteria at the bond level, which readily enables the simulation of brittle and ductile fracture. Discretizing the model with asymptotically compatible meshfree approximation provides a scheme which converges to the classical KL shell model while providing an accurate and flexible framework for treating fracture. A wide range of numerical examples, ranging from elastostatics to problems involving plasticity, fracture, and fragmentation, are conducted to validate the accuracy, convergence, and robustness of the developed PD thin-shell formulation. It is also worth noting that the present method naturally enables the discretization of a shell theory requiring higher-order smoothness on a completely unstructured surface mesh.
... A comprehensive set of numerical evaluations are 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), 10 TempestRemap (Ullrich and Taylor, 2015), Generalized Moving-Least-Squares (GMLS) (Trask and Kuberry, 2020) with postprocessing 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 the challenging, realistic field data on both uniform and regionally refined mesh cases. ...
... The demonstrated techniques used in high-order interpolators 15 use spline quasi-interpolants (de Boor, 1990), bi-cubic splines (Hanke et al., 2016;Craig et al., 2017), the standard radial-basis function spaces (Flyer and Wright, 2007;Bungartz et al., 2016), the moving least-squares (MLS) method (Lancaster and Salkauskas, 1981), and its variants such as the modified MLS (MMLS) (Joldes et al., 2015;Slattery, 2016), which originate from high-order reconstruction methods. Recent extensions to MLS for producing efficient high-order remap involve reconstructing locally the manifold geometry from a point set representation, and then generating a compact stencil in the local coordinate 20 chart (Liang and Zhao, 2013;Suchde and Kuhnert, 2019;Trask and Kuberry, 2020;Gross et al., 2020). ...
... However, in this work, reconstruction of functions sampled on a manifold permits generating a compact stencil in a local 25 coordinate chart, which is one dimension smaller (Liang and Zhao, 2013;Suchde and Kuhnert, 2019;Trask and Kuberry, 2020;Gross et al., 2020). The savings in net computational floating-point operations (flops) is a factor of p 3 in R 2 , as compared to a traditional basis in R 3 , where p is the order of the basis. ...
Preprint
Full-text available
Strongly coupled nonlinear phenomena such as those described by Earth System Models (ESM) are composed of multiple component models with independent mesh topologies and scalable numerical solvers. A common operation in ESM is to remap or interpolate results from one component's computational mesh to another, 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 ESM. 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 are 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, TempestRemap, Generalized Moving-Least-Squares (GMLS) with post-processing filters, and Weighted-Least-Squares Essentially Non-oscillatory Remap (WLS-ENOR) method. 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 the 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.
... In the broader meshfree discretization literature, a number of works have developed discretizations of general surface PDE [45,111,62,96,104,105,81,110,95,73,43,88,68]. Such schemes treat the manifold extrinsically, working in ambient space and projecting to manifold, or intrinsically, using compact reconstructions of the manifold to obtain local coordinates. ...
... Such schemes treat the manifold extrinsically, working in ambient space and projecting to manifold, or intrinsically, using compact reconstructions of the manifold to obtain local coordinates. The latter generally apply polynomial reconstruction to obtain estimates of the metric tensor necessary for computing surface differential operators [63,51,4,45,111,62]. In contrast, by working in the reference configuration, the linear PCA reconstruction used in the current approach provides high-order treatment of surface curvature, avoiding the need for either high-order polynomial reconstruction or use of greater than C 0 continuity shape functions. ...
... We start with a meshfree, local parameterization of the shell mid-surface. We employ the approach detailed in [111] to define a local parameterization of a point cloud without resorting to using a global parametric map. Considering the undeformed configuration of a mid-surface B 2D , we approximate a tangent plane at each surface mesh point P ∈ B 2D using Principal Component Analysis (PCA) [118]. ...
Preprint
Full-text available
We present a comprehensive rotation-free Kirchhoff-Love (KL) shell formulation for peridynam-ics (PD) that is capable of modeling large elasto-plastic deformations and fracture in thin-walled structures. To remove the need for a predefined global parametric domain, Principal Component Analysis is employed in a meshfree setting to develop a local parameterization of the shell mid-surface. The KL shell kinematics is utilized to develop a correspondence-based PD formulation. A bond-stabilization technique is employed to naturally achieve stability of the discrete solution. Only the mid-surface velocity degrees of freedom are used in the governing thin-shell equations. 3D rate-form material models are employed to enable simulating a wide range of material behavior. A bond-associative damage correspondence modeling approach is adopted to use classical failure criteria at the bond level, which readily enables the simulation of brittle and ductile fracture. Discretizing the model with asymptotically compatible meshfree approximation provides a scheme which converges to the classical KL shell model while providing an accurate and flexible framework for treating fracture. A wide range of numerical examples, ranging from elastostatics to problems involving plasticity, fracture, and fragmentation, are conducted to validate the accuracy, convergence, and robustness of the developed PD thin-shell formulation. It is also worth noting that the present method naturally enables the discretization of a shell theory requiring higher-order smoothness on a completely unstructured surface mesh.
... The demonstrated techniques used in high-order interpolators use spline quasiinterpolants (de Boor, 1990), bi-cubic splines (Hanke et al., 2016;Craig et al., 2017), the standard radial-basis function spaces (Flyer and Wright, 2007;Bungartz et al., 2016), the moving least squares (MLS) method (Lancaster and Salkauskas, 1981), and MLS variants such as the modified MLS (MMLS) (Joldes et al., 2015;Slattery, 2016), which originate from high-order reconstruction methods. Recent extensions to MLS for producing efficient high-order remap involve locally reconstructing the manifold geometry from a point set representation and then generating a compact stencil in the local coordinate chart (Liang and Zhao, 2013;Suchde and Kuhnert, 2019;Trask and Kuberry, 2020;Gross et al., 2020). ...
... Traditionally, GMLS uses a basis that is defined as a function of the spatial dimension from which a point cloud is sampled. However, in this work, reconstruction of functions sampled on a manifold permits generating a compact stencil in a local coordinate chart, which is one dimension smaller (Liang and Zhao, 2013;Suchde and Kuhnert, 2019;Trask and Kuberry, 2020;Gross et al., 2020). The savings in net computational floating-point operations (flops) is a factor of p 3 in R 2 compared to a traditional basis in R 3 , where p is the order of the basis. ...
Article
Full-text available
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.
... These techniques include surface finite element (SFE) [16], embedded finite element (EFE) [8,33], and closest point (CP) [29] methods. More recently, various meshfree (or meshless) methods have also been developed for PDEs on general surfaces that use a local stencil approach, including radial basis function-finite differences (RBF-FD) [2,26,37,41,42,50], generalized finite differences (GFD) [45], and generalized moving least squares (GMLS) [20,27,46]. These methods can be applied for surfaces represented only by point clouds and do not require a surface triangulation like SFE methods or a level-set representation of the surface like EFE methods. ...
... 2. Localized meshfree discretizations. Several localized meshfree methods have been developed for approximating the solution of (1.1), e.g., [2,26,27,37,41,42,46]. For the sake of brevity, we limit the focus of this study to two localized meshfree methods: polyharmonic spline (PHS)based RBF-FD with polynomials and GFD. ...
Preprint
We develop a new meshfree geometric multilevel (MGM) method for solving linear systems that arise from discretizing elliptic PDEs on surfaces represented by point clouds. The method uses a Poisson disk sampling-type technique for coarsening the point clouds and new meshfree restriction/interpolation operators based on polyharmonic splines for transferring information between the coarsened point clouds. These are then combined with standard smoothing and operator coarsening methods in a V-cycle iteration. MGM is applicable to discretizations of elliptic PDEs based on various localized meshfree methods, including RBF finite differences (RBF-FD) and generalized finite differences (GFD). We test MGM both as a standalone solver and preconditioner for Krylov subspace methods on several test problems using RBF-FD and GFD, and numerically analyze convergence rates, efficiency, and scaling with increasing point cloud sizes. We also perform a side-by-side comparison to algebraic multigrid (AMG) methods for solving the same systems. Finally, we further demonstrate the effectiveness of MGM by applying it to three challenging applications on complicated surfaces: pattern formation, surface harmonics, and geodesic distance.
... In the Euclidean setting, this has allowed for stable GMLS discretizations of Darcy flow in R d [95], Stokes flow in R d [104], and fluid-structure interactions occurring in suspension flow [107]. In the recent work [111], it has been shown that the scheme developed by Liang et al. [72] to discretize the Laplace-Beltrami operator on manifolds admits an interpretation as a GMLS approximation. This unification enables extensions of the compatible staggered approach for Darcy in R d [95] to the manifold setting [111]. ...
... In the recent work [111], it has been shown that the scheme developed by Liang et al. [72] to discretize the Laplace-Beltrami operator on manifolds admits an interpretation as a GMLS approximation. This unification enables extensions of the compatible staggered approach for Darcy in R d [95] to the manifold setting [111]. ...
Article
Full-text available
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.
... In the Euclidean setting, this has allowed for stable GMLS discretizations of Darcy flow in R d [97] and Stokes flow in R d [95], and to adaptively study fluid-structure interactions occurring in suspension flow [50]. In a recent work [94], we have shown that the scheme developed by Liang et al. [60] to discretize the Laplace-Beltrami operator on manifolds admits an interpretation as a GMLS approximation. This unification enabled extensions of our compatible staggered approach for Darcy in R d [97] to the manifold setting [94]. ...
... In a recent work [94], we have shown that the scheme developed by Liang et al. [60] to discretize the Laplace-Beltrami operator on manifolds admits an interpretation as a GMLS approximation. This unification enabled extensions of our compatible staggered approach for Darcy in R d [97] to the manifold setting [94]. ...
Preprint
Full-text available
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 a splitting scheme of the fourth-order governing equations in terms of two second-order elliptic operators. We show our methods have high-order convergence rates for the metric and other geometric quantities of the manifold, for the truncation errors of exterior calculus operators, and for the solution errors of the Stokes problem for hydrodynamic flows on curved surfaces. Our approaches also may be utilized to develop high-order solvers for other scalar-valued and vector-valued problems on manifolds.
... Methods with rigorous convergence results include the wide stencil schemes for Hamilton-Jacobi equations and elliptic PDEs [62], which were originally defined on regular grids and have subsequently been extended to unstructured point clouds [43,47], and the point integral method [55]. Other works without convergence guarantees include upwind schemes for Hamilton-Jacobi equations on unstructured meshes [68], mesh-free generalized finite difference methods [71,72], least squares manifold approximation methods [56,75,78], the local mesh method [53], radial basis function methods [45,48,63,64], and a recent approach using graph Laplacians and deep learning [57]. A general survey of meshfree methods in PDEs is given in [28]. ...
Article
Full-text available
We investigate identifying the boundary of a domain from sample points in the domain. We introduce new estimators for the normal vector to the boundary, distance of a point to the boundary, and a test for whether a point lies within a boundary strip. The estimators can be efficiently computed and are more accurate than the ones present in the literature. We provide rigorous error estimates for the estimators. Furthermore we use the detected boundary points to solve boundary-value problems for PDE on point clouds. We prove error estimates for the Laplace and eikonal equations on point clouds. Finally we provide a range of numerical experiments illustrating the performance of our boundary estimators, applications to PDE on point clouds, and tests on image data sets.
... Methods with rigorous convergence results include the wide stencil schemes for Hamilton-Jacobi equations and elliptic PDEs [60], which were originally defined on regular grids and have subsequently been extended to unstructured point clouds [41,45], and the point integral method [53]. Other works without convergence guarantees include upwind schemes for Hamilton-Jacobi equations on unstructured meshes [65], mesh-free generalized finite difference methods [68,69], least squares manifold approximation methods [54,72,75], the local mesh method [51], radial basis function methods [43,46,61,62], and a recent approach using graph Laplacians and deep learning [55]. A general survey of meshfree methods in PDEs is given in [26]. ...
Preprint
We investigate identifying the boundary of a domain from sample points in the domain. We introduce new estimators for the normal vector to the boundary, distance of a point to the boundary, and a test for whether a point lies within a boundary strip. The estimators can be efficiently computed and are more accurate than the ones present in the literature. We provide rigorous error estimates for the estimators. Furthermore we use the detected boundary points to solve boundary-value problems for PDE on point clouds. We prove error estimates for the Laplace and eikonal equations on point clouds. Finally we provide a range of numerical experiments illustrating the performance of our boundary estimators, applications to PDE on point clouds, and tests on image data sets.
... We then perform studies showing how FPTs are influenced by the surface geometry, drift dynamics, and spatially dependent diffusivities. methods for solving PDEs on surfaces related to [19,27,45,52,53]. We focus particularly on the mean first passage times given by the special case f = 0, g = 1 with u(x) = E x [τ ]. ...
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Full-text available
We develop numerical methods for computing statistics of stochastic processes on surfaces of general shape with drift-diffusion dynamics $d{X}_t = a({X}_t)dt + {b}({X}_t)d{W}_t$. We consider on a surface domain $\Omega$ the statistics $u(\mathbf{x}) = \mathbb{E}^{\mathbf{x}}\left[\int_0^\tau g(X_t)dt \right] + \mathbb{E}^{\mathbf{x}}\left[f(X_\tau)\right]$ with the exit stopping time $\tau = \inf_t \{t > 0 \; |\; X_t \not\in \Omega\}$. Using Dynkin's formula, we compute statistics by developing high-order Generalized Moving Least Squares (GMLS) solvers for the associated surface PDE boundary-value problems. We focus particularly on the mean First Passage Times (FPTs) given by the special case $f = 0,\, g = 1$ with $u(\mathbf{x}) = \mathbb{E}^{\mathbf{x}}\left[\tau\right]$. 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 FPTs are influenced by the surface geometry, drift dynamics, and spatially dependent diffusivities.
... Trask et al. [35] proposed a high-order staggered meshless method, which combines a topological gradient operator on the local primal grid with a GMLS approximation of the divergence on the local dual grid, to solve problems with discontinuous coefficients. Trask et al. [34] presented a compatible meshfree discretization of elliptic problems on manifolds. Hu et al. [19] proposed an a posteriori adaptive refinement strategy in the context of GMLS to simulate the flows of colloid suspensions. ...
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In this article a generalized finite difference method (GFDM), which is a meshless method based on Taylor series expansions and weighted moving least squares, is proposed to solve the elliptic interface problem. This method turns the original elliptic interface problem to be two coupled elliptic non-interface subproblems. The solutions are found by solving coupled elliptic subproblems with sparse coefficient matrix, which significantly improves the efficiency for the interface problem, especially for the complex geometric interface. Furthermore, based on the key idea of GFDM which can approximate the derivatives of unknown variables by linear summation of nearby nodal value, we further develop the GFDM to deal with the elliptic problem with the jump interface condition which related to the derivative of solution on the interface boundary. Four numerical examples are provided to illustrate the features of the proposed method, including the acceptable accuracy and the efficiency.
Article
This paper presents an efficient collocation method which combines the generalized finite difference method (GFDM) with the Krylov deferred correction (KDC) method for the long-time simulation of heat and mass transport on evolving surfaces. The KDC method utilizes a pseudo-spectral-type temporal collocation formulation to discretize the time-dependent surface heat and mass transport equation in each time marching step, where the time derivatives at the collocation points are introduced as the new unknown variables. A low-order time marching scheme is then applied as an effective preconditioner in the Jacobian-Free Newton-Krylov framework to decouple the spatial surface PDEs at different collocation nodes. Each decoupled surface PDE is then solved by the meshless GFDM, where both the continuous-form evolving surfaces defined by parametric equations and discretized-form evolving surfaces composed of point clouds are considered in the GFDM spatial discretization. Numerical experiments show that the combined GFDM-KDC solver is a promising numerical scheme for long-time evolution simulation of heat and mass transport on intractable evolving surfaces.
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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.
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In this work, we propose an efficient meshfree method based on Pascal polynomials and mutiple-scale approach for numerical solutions of two-dimensional (2-D) and three-dimensional (3-D) elliptic interface problems which may have discontinuous coefficients and curved interfaces with or without sharp corners. The proposed method uses Pascal polynomials as basis functions and utilizes multiple-scale approach for stabilizing numerical solutions. It is a well known fact that using polynomial basis without any modifications to obtain numerical solutions of partial differential equations may be peculiar owing to ill conditioned resultant coefficent matrix which is formed after process of discretization. Hence, as a remedy to the highly ill-conditioned coefficient matrix we employ multiple-scale approach. The main idea behind the multiple-scale approach is to make norm of all columns of resultant coefficient matrix equal to each other. The proposed method is a truly strong-form meshfree method since we do not need any mesh or integration process in problem domain, these features makes the implementation of the method very simple in computer environment. The efficiency of the proposed method is tested by some test problems which may have smooth interface or interface with sharp corners. Stability of the proposed method is investigated numerically in the presence of noise effect. Further, to show accuracy of the proposed method we present some comparisons with available numerical methods in literature, such as direct meshless local Petrov-Galerkin method, matched interface and boundary method, spectral element method and some meshless methods based on radial basis functions. The obtained numerical results and their comparisons confirm applicability of the proposed method for 2-D and 3-D steady state elliptic interface problems.
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We apply a local maximum entropy (LME) approximation scheme to the fourth order phase-field model for the traditional Cahn-Hilliard theory. The discretization of the Cahn-Hilliard equation by Galerkin method requires at least $C^1$-continuous basis functions. This requirement can be fulfilled by using LME shape functions which are $C^\infty$-continuous. In this case, the primal variational formulations of the fourth-order partial differential equation is well defined and integrable. Hence, there is no need to split the fourth-order partial differential equation into two second-order partial differential equations; this splitting scheme is a common practice in mixed finite element formulations with $C^0$-continuous Lagrange shape functions. Furthermore, we use a general and simple numerical method such as statistical manifold learning techniques that allows dealing with general point set surfaces avoiding a global parametrization, which can be applied to tackle surfaces of complex geometry and topology, and to solve Cahn-Hilliard equation on general surfaces. Finally, the flexibility and robustness of the presented methodology is demonstrated for several representative numerical examples.
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Partial differential equations (PDEs) on surfaces appear in many applications throughout the natural and applied sciences. The classical closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) is an embedding method for solving PDEs on surfaces using standard finite difference schemes. In this paper, we formulate an explicit closest point method using finite difference schemes derived from radial basis functions (RBF- FD). Unlike the orthogonal gradients method (Piret, J. Comput. Phys. 231(14):4662-4675, [2012]), our proposed method uses RBF centers on regular grid nodes. This formulation not only reduces the computational cost but also avoids the ill-conditioning from point clustering on the surface and is more natural to couple with a grid based manifold evolution algorithm (Leung and Zhao, J. Comput. Phys. 228(8):2993-3024, [2009]). When compared to the standard finite difference discretization of the closest point method, the proposed method requires a smaller computational domain surrounding the surface, resulting in a decrease in the number of sampling points on the surface. In addition, higher-order schemes can easily be constructed by increasing the number of points in the RBF-FD stencil. Applications to a variety of examples are provided to illustrate the numerical convergence of the method.
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We present three new semi-Lagrangian methods based on radial basis function (RBF) interpolation for numerically simulating transport on a sphere. The methods are mesh-free and are formulated entirely in Cartesian coordinates, thus avoiding any irregular clustering of nodes at artificial boundaries on the sphere and naturally bypassing any apparent artificial singularities associated with surface-based coordinate systems. For problems involving tracer transport in a given velocity field, the semi-Lagrangian framework allows these new methods to avoid the use of any stabilization terms (such as hyperviscosity) during time-integration, thus reducing the number of parameters that have to be tuned. The three new methods are based on interpolation using 1) global RBFs, 2) local RBF stencils, and 3) RBF partition of unity. For the latter two of these methods, we find that it is crucial to include some low degree spherical harmonics in the interpolants. Standard test cases consisting of solid body rotation and deformational flow are used to compare and contrast the methods in terms of their accuracy, efficiency, conservation properties, and dissipation/dispersion errors. For global RBFs, spectral spatial convergence is observed for smooth solutions on quasi-uniform nodes, while high-order accuracy is observed for the local RBF stencil and partition of unity approaches.
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The IMA Hot Topics workshop on compatible spatialdiscretizations was held May 11-15, 2004 at the University of Minnesota. The purpose of the workshop was to bring together scientists at the forefront of the research in the numerical solution of PDEs to discuss recent advances and novel applications of geometrical and homological approaches to discretization. This volume contains original contributions based on the material presented at the workshop. A unique feature of the collection is the inclusion of work that is representative of the recent developments in compatible discretizations across a wide spectrum of disciplines in computational science. Compatible spatial discretizations are those that inherit or mimic fundamental properties of the PDE such as topology, conservation, symmetries, and positivity structures and maximum principles. The papers in the volume offer a snapshot of the current trends and developments in compatible spatial discretizations. The reader will find valuable insights on spatial compatibility from several different perspectives and important examples of applications compatible discretizations in computational electromagnetics, geosciences, linear elasticity, eigenvalue approximations and MHD. The contributions collected in this volume will help to elucidate relations between different methods and concepts and to generally advance our understanding of compatible spatial discretizations for PDEs. Abstracts and presentation slides from the workshop can be accessed at http://www.ima.umn.edu/talks/workshops/5-11-15.2004/.
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The Moving Least Squares (MLS) method provides an approxima-tio u of a function u based solely on values u(x j) of u on scattered "meshless" nodes x j . Derivatives of u are usually approximated by derivatives o u. In contrast to this, we directly estimate derivatives of u from the data, without any detour via derivatives o u. This is a generalized Moving Least Squares technique, and we prove that it produces diffuse derivatives as introduced by Nyroles et. al. in 1992. Consequently, these turn out to be efficient direct estimates of the true derivatives, without anything "diffuse" about them, and we prove optimal rates of convergence towards the true derivatives. Numerical examples confirm this, and we finally show how the use of shifted and scaled polynomials as basis functions in the generalized and standard MLS approxi-mation stabilizes the algorithm.
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The Trilinos Project is an effort to facilitate the design, development, integration, and ongoing support of mathematical software libraries within an object-oriented framework for the solution of large-scale, complex multiphysics engineering and scientific problems. Trilinos addresses two fundamental issues of developing software for these problems: (i) providing a streamlined process and set of tools for development of new algorithmic implementations and (ii) promoting interoperability of independently developed software.Trilinos uses a two-level software structure designed around collections of packages. A Trilinos package is an integral unit usually developed by a small team of experts in a particular algorithms area such as algebraic preconditioners, nonlinear solvers, etc. Packages exist underneath the Trilinos top level, which provides a common look-and-feel, including configuration, documentation, licensing, and bug-tracking.Here we present the overall Trilinos design, describing our use of abstract interfaces and default concrete implementations. We discuss the services that Trilinos provides to a prospective package and how these services are used by various packages. We also illustrate how packages can be combined to rapidly develop new algorithms. Finally, we discuss how Trilinos facilitates high-quality software engineering practices that are increasingly required from simulation software.
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Least-squares finite element methods are an attractive class of methods for the numerical solution of partial differential equations. They are motivated by the desire to recover, in general settings, the advantageous features of Rayleigh�Ritz methods such as the avoidance of discrete compatibility conditions and the production of symmetric and positive definite discrete systems. The methods are based on the minimization of convex functionals that are constructed from equation residuals. This paper focuses on theoretical and practical aspects of least-square finite element methods and includes discussions of what issues enter into their construction, analysis, and performance. It also includes a discussion of some open problems.
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The Compadre Toolkit provides a performance portable solution for the parallel evaluation of computationally dense kernels. The toolkit specifically targets the Generalized Moving Least Squares (GMLS) approach, which requires the inversion of small dense matrices. The result is a set of weights that provide the information needed for remap or entries that constitute the rows of some globally sparse matrix. This toolkit focuses on the 'on-node' aspects of meshless PDE solution and remap, namely the parallel construction of small dense matrices and their inversion. What it does not provide is the tools for managing fields, inverting globally sparse matrices, or neighbor search that requires orchestration over many MPI processes. This toolkit is designed to be easily dropped-in to an existing MPI (or serial) based framework for PDE solution or remap, with minimal dependencies (Kokkos and either Cuda Toolkit or LAPACK).
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Part I: Equations of motion and some basic ideas on discretizations.- Pat II. Conservation laws, finite-volume methods, remapping techniques and spherical grids.- Part III. Some aspects of atmospheric dynamical cores
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1. Applications and motivations 2. Hear spaces and multivariate polynomials 3. Local polynomial reproduction 4. Moving least squares 5. Auxiliary tools from analysis and measure theory 6. Positive definite functions 7. Completely monotine functions 8. Conditionally positive definite functions 9. Compactly supported functions 10. Native spaces 11. Error estimates for radial basis function interpolation 12. Stability 13. Optimal recovery 14. Data structures 15. Numerical methods 16. Generalised interpolation 17. Interpolation on spheres and other manifolds.
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The current paper establishes the computational efficiency and accuracy of the RBF-FD method for large-scale geoscience modeling with comparisons to state-of-the-art methods as high-order discontinuous Galerkin and spherical harmonics, the latter using expansions with close to 300,000 bases. The test cases are demanding fluid flow problems on the sphere that exhibit numerical challenges, such as Gibbs phenomena, sharp gradients, and complex vortical dynamics with rapid energy transfer from large to small scales over short time periods. The computations were possible as well as very competitive due to the implementation of hyperviscosity on large RBF stencil sizes (corresponding roughly to 6th to 9th order methods) with up to O(10 5 ) nodes on the sphere. The RBF-FD method scaled as O(N) per time step, where N is the total number of nodes on the sphere. In Appendix A, guidelines are given on how to chose parameters when using RBF-FD to solve hyperbolic PDEs.
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We present a finite element method for the Stokes equations involving two immiscible incompressible fluids with different viscosities and with surface tension. The interface separating the two fluids does not need to align with the mesh. We propose a Nitsche formulation which allows for discontinuities along the interface with optimal a priori error estimates. A stabilization procedure is included which ensures that the method produces a well conditioned stiffness matrix independent of the location of the interface.
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A meshless approach to the analysis of arbitrary Kirchhoff shells by the Element-Free Galerkin (EFG) method is presented. The shell theory used is geometrically exact and can be applied to deep shells. The method is based on moving least squares approximant. The method is meshless, which means that the discretization is independent of the geometric subdivision into “finite elements”. The satisfaction of the C1 continuity requirements is easily met by EFG since it requires only C1 weights; therefore, it is not necessary to resort to Mindlin-Reissner theory or to devices such as discrete Kirchhoff theory. The requirements of consistency are met by the use of a polynomial basis of quadratic or higher order. A subdivision similar to finite elements is used to provide a background mesh for numerical integration. The essential boundary conditions are enforced by Lagrange multipliers. Membrane locking, which is due to different approximation order for transverse and membrane displacements, is removed by using larger domains of influence with the quadratic basis, and by using quartic polynomial basis, which can prevent membrane locking completely. It is shown on the obstacle course for shells that the present technique performs well.
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We develop new stabilized mixed finite element methods for Darcy flow. Stability and an a priori error estimate in the “stability norm” are established. A wide variety of convergent finite elements present themselves, unlike the classical Galerkin formulation which requires highly specialized elements. An interesting feature of the formulation is that there are no mesh-dependent parameters. Numerical tests confirm the theoretical results.
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A derivation of the convective‐diffusion equation for transport of a scalar quantity, e.g., surfactant, along a deforming interface is outlined. The direct contribution of interface deformation, giving rise to concentration variations as a result of local changes in interfacial area, is shown explicitly in a simple manner.
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