Partitioned methods allow one to build a simulation capability for coupled problems by reusing existing single-component codes. In so doing, partitioned methods can shorten code development and validation times for multiphysics and multiscale applications. In this work, we consider a scenario in which one or more of the “codes” being coupled are projection-based reduced order models (ROMs), introduced to lower the computational cost associated with a particular component. We simulate this scenario by considering a model interface problem that is discretized independently on two non-overlapping subdomains. Here we then formulate a partitioned scheme for this problem that allows the coupling between a ROM “code” for one of the subdomains with a finite element model (FEM) or ROM “code” for the other subdomain. The ROM “codes” are constructed by performing proper orthogonal decomposition (POD) on a snapshot ensemble to obtain a low-dimensional reduced order basis, followed by a Galerkin projection onto this basis. The ROM and/or FEM “codes” on each subdomain are then coupled using a Lagrange multiplier representing the interface flux. To partition the resulting monolithic problem, we first eliminate the flux through a dual Schur complement. Application of an explicit time integration scheme to the transformed monolithic problem decouples the subdomain equations, allowing their independent solution for the next time step. We show numerical results that demonstrate the proposed method’s efficacy in achieving both ROM-FEM and ROM-ROM coupling.
Nonlocal models provide a much-needed predictive capability for important Sandia mission applications, ranging from fracture mechanics for nuclear components to subsurface flow for nuclear waste disposal, where traditional partial differential equations (PDEs) models fail to capture effects due to long-range forces at the microscale and mesoscale. However, utilization of this capability is seriously compromised by the lack of a rigorous nonlocal interface theory, required for both application and efficient solution of nonlocal models. To unlock the full potential of nonlocal modeling we developed a mathematically rigorous and physically consistent interface theory and demonstrate its scope in mission-relevant exemplar problems.
Mahadevan, Vijay S.; Guerra, Jorge E.; Jiao, Xiangmin; Kuberry, Paul A.; Li, Yipeng; Ullrich, Paul; Marsico, David; Jacob, Robert; Bochev, Pavel B.; Jones, Philip
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 , TempestRemap , generalized moving least squares (GMLS) with post-processing filters, and WLS-ENOR . 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.
We develop and analyze an optimization-based method for the coupling of a static peridynamic (PD) model and a static classical elasticity model. The approach formulates the coupling as a control problem in which the states are the solutions of the PD and classical equations, the objective is to minimize their mismatch on an overlap of the PD and classical domains, and the controls are virtual volume constraints and boundary conditions applied at the local-nonlocal interface. Our numerical tests performed on three-dimensional geometries illustrate the consistency and accuracy of our method, its numerical convergence, and its applicability to realistic engineering geometries. We demonstrate the coupling strategy as a means to reduce computational expense by confining the nonlocal model to a subdomain of interest, and as a means to transmit local (e.g., traction) boundary conditions applied at a surface to a nonlocal model in the bulk of the domain.
We present an optimization-based coupling method for local and nonlocal continuum models. Our approach couches the coupling of the models into a control problem where the states are the solutions of the nonlocal and local equations, the objective is to minimize their mismatch on the overlap of the local and nonlocal problem domains, and the virtual controls are the nonlocal volume constraint and the local boundary condition. We present the method in the context of Local-to-Nonlocal diffusion coupling. Numerical examples illustrate the theoretical properties of the approach.
Component coupling is a crucial part of climate models, such as DOE's E3SM (Caldwell et al., 2019). A common coupling strategy in climate models is for their components to exchange flux data from the previous time-step. This approach effectively performs a single step of an iterative solution method for the monolithic coupled system, which may lead to instabilities and loss of accuracy. In this paper we formulate an Interface-Flux-Recovery (IFR) coupling method which improves upon the conventional coupling techniques in climate models. IFR starts from a monolithic formulation of the coupled discrete problem and then uses a Schur complement to obtain an accurate approximation of the flux across the interface between the model components. This decouples the individual components and allows one to solve them independently by using schemes that are optimized for each component. To demonstrate the feasibility of the method, we apply IFR to a simplified ocean–atmosphere model for heat-exchange coupled through the so-called bulk condition, common in ocean–atmosphere systems. We then solve this model on matching and non-matching grids to estimate numerically the convergence rates of the IFR coupling scheme.
This paper continues our efforts to exploit optimization and control ideas as a common foundation for the development of property-preserving numerical methods. Here we focus on a class of scalar advection equations whose solutions have fixed mass in a given Eulerian region and constant bounds in any Lagrangian volume. Our approach separates discretization of the equations from the preservation of their solution properties by treating the latter as optimization constraints. This relieves the discretization process from having to comply with additional restrictions and makes stability and accuracy the sole considerations in its design. A property-preserving solution is then sought as a state that minimizes the distance to an optimally accurate but not property-preserving target solution computed by the scheme, subject to constraints enforcing discrete proxies of the desired properties. We consider two such formulations in which the optimization variables are given by the nodal solution values and suitably defined nodal fluxes, respectively. A key result of the paper reveals that a standard Algebraic Flux Correction (AFC) scheme is a modified version of the second formulation obtained by shrinking its feasible set to a hypercube. We conclude with numerical studies illustrating the optimization-based formulations and comparing them with AFC.
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.
Proceedings of the 6th European Conference on Computational Mechanics: Solids, Structures and Coupled Problems, ECCM 2018 and 7th European Conference on Computational Fluid Dynamics, ECFD 2018
We present an optimization approach with two controls for coupling elliptic partial differential equations posed on subdomains sharing an interface that is discretized independently on each subdomain, introducing gaps and overlaps. We use two virtual Neumann controls, one defined on each discrete interface, thereby eliminating the need for a virtual common refinement interface mesh. Global flux conservation is achieved by including the square of the difference of the total flux on each interface in the objective. We use Generalized Moving Least Squares (GMLS) reconstruction to evaluate and compare the subdomain solution and gradients at quadrature points used in the cost functional. The resulting method recovers globally linear solutions and shows optimal L2-norm and H1-norm convergence.
In most finite element methods the mesh is used to both represent the domain and to define the finite element basis. As a result the quality of such methods is tied to the quality of the mesh and may suffer when the latter deteriorates. This paper formulates an alternative approach, which separates the discretization of the domain, i.e., the meshing, from the discretization of the PDE. The latter is accomplished by extending the Generalized Moving Least-Squares (GMLS) regression technique to approximation of bilinear forms and using the mesh only for the integration of the GMLS polynomial basis. Our approach yields a non-conforming discretization of the weak equations that can be handled by standard discontinuous Galerkin or interior penalty terms.
We present a numerical method for synchronous and concurrent solution of transient elastodynamics problem where the computational domain is divided into subdomains that may reside on separate computational platforms. This work employs the variational multiscale discontinuous Galerkin (VMDG) method to develop interdomain transmission conditions for transient problems. The fine-scale modeling concept leads to variationally consistent coupling terms at the common interfaces. The method admits a large class of time discretization schemes, and decoupling of the solution for each subdomain is achieved by selecting any explicit algorithm. Numerical tests with a manufactured solution problem show optimal convergence rates. The energy history in a free vibration problem is in agreement with that of the solution from a monolithic computational domain.
Proceedings of the 6th European Conference on Computational Mechanics: Solids, Structures and Coupled Problems, ECCM 2018 and 7th European Conference on Computational Fluid Dynamics, ECFD 2018
As the mean time between failures on the future high-performance computing platforms is expected to decrease to just a few minutes, the development of “smart”, property-preserving checkpointing schemes becomes imperative to avoid dramatic decreases in application utilization. In this paper we formulate a generic optimization-based approach for fault-tolerant computations, which separates property preservation from the compression and recovery stages of the checkpointing processes. We then specialize the approach to obtain a fault recovery procedure for a model scalar transport equation, which preserves local solution bounds and total mass. Numerical examples showing solution recovery from a corrupted application state for three different failure modes illustrate the potential of the approach.
Compact semiconductor device models are essential for efficiently designing and analyzing large circuits. However, traditional compact model development requires a large amount of manual effort and can span many years. Moreover, inclusion of new physics (e.g., radiation effects) into an existing model is not trivial and may require redevelopment from scratch. Machine Learning (ML) techniques have the potential to automate and significantly speed up the development of compact models. In addition, ML provides a range of modeling options that can be used to develop hierarchies of compact models tailored to specific circuit design stages. In this paper, we explore three such options: (1) table-based interpolation, (2) Generalized Moving Least-Squares, and (3) feedforward Deep Neural Networks, to develop compact models for a p-n junction diode. We evaluate the performance of these "data-driven" compact models by (1) comparing their voltage-current characteristics against laboratory data, and (2) building a bridge rectifier circuit using these devices, predicting the circuit's behavior using SPICE-like circuit simulations, and then comparing these predictions against laboratory measurements of the same circuit.
This report summarizes the work performed under a three year LDRD project aiming to develop mathematical and software foundations for compatible meshfree and particle discretizations. We review major technical accomplishments and project metrics such as publications, conference and colloquia presentations and organization of special sessions and minisimposia. The report concludes with a brief summary of ongoing projects and collaborations that utilize the products of this work.
Traditional explicit partitioned schemes exchange boundary conditions between subdomains and can be related to iterative solution methods for the coupled problem. As a result, these schemes may require multiple subdomain solves, acceleration techniques, or optimized transmission conditions to achieve sufficient accuracy and/or stability. We present a new synchronous partitioned method derived from a well-posed mixed finite element formulation of the coupled problem. We transform the resulting Differential Algebraic Equation (DAE) to a Hessenberg index-1 form in which the algebraic equation defines the Lagrange multiplier as an implicit function of the states. Using this fact we eliminate the multiplier and reduce the DAE to a system of explicit ODEs for the states. Explicit time integration both discretizes this system in time and decouples its equations. As a result, the temporal accuracy and stability of our formulation are governed solely by the accuracy and stability of the explicit scheme employed and are not subject to additional stability considerations as in traditional partitioned schemes. We establish sufficient conditions for the formulation to be well-posed and prove that classical mortar finite elements on the interface are a stable choice for the Lagrange multiplier. We show that in this case the condition number of the Schur complement involved in the elimination of the multiplier is bounded by a constant. The paper concludes with numerical examples illustrating the approach for two different interface problems.
Nonlocal continuum theories for mechanics can capture strong nonlocal effects due to long-range forces in their governing equations. When these effects cannot be neglected, nonlocal models are more accurate than partial differential equations (PDEs); however, the accuracy comes at the price of a prohibitive computational cost, making local-to-nonlocal (LtN) coupling strategies mandatory. In this chapter, we review the state of the art of LtN methods where the efficiency of PDEs is combined with the accuracy of nonlocal models. Then, we focus on optimization-based coupling strategies that couch the coupling of the models into a control problem where the states are the solutions of the nonlocal and local equations, the objective is to minimize their mismatch on the overlap of the local and nonlocal problem domains, and the virtual controls are the nonlocal volume constraint and the local boundary condition. The strategy is described in the context of nonlocal and local elasticity and illustrated by numerical tests on three-dimensional realistic geometries. Additional numerical tests also prove the consistency of the method via patch tests.
A discrete De Rham complex enables compatible, structure-preserving discretizations for a broad range of partial differential equations problems. Such discretizations can correctly reproduce the physics of interface problems, provided the grid conforms to the interface. However, large deformations, complex geometries, and evolving interfaces makes generation of such grids difficult. We develop and demonstrate two formally equivalent approaches that, for a given background mesh, dynamically construct an interface-conforming discrete De Rham complex. Both approaches start by dividing cut elements into interface-conforming subelements but differ in how they build the finite element basis on these subelements. The first approach discards the existing non-conforming basis of the parent element and replaces it by a dynamic set of degrees of freedom of the same kind. The second approach defines the interface-conforming degrees of freedom on the subelements as superpositions of the basis functions of the parent element. These approaches generalize the Conformal Decomposition Finite Element Method (CDFEM) and the extended finite element method with algebraic constraints (XFEM-AC), respectively, across the De Rham complex.
Independent meshing of subdomains separated by an interface can lead to spatially non-coincident discrete interfaces. We present an optimization-based coupling method for such problems, which does not require a common mesh refinement of the interface, has optimal H1 convergence rates, and passes a patch test. The method minimizes the mismatch of the state and normal stress extensions on discrete interfaces subject to the subdomain equations, while interface “fluxes” provide virtual Neumann controls.
In this work we present a computational capability featuring a hierarchy of models with different fidelities for the solution of electrokinetics problems at the micro-/nano-scale. A multifidelity approach allows the selection of the most appropriate model, in terms of accuracy and computational cost, for the particular application at hand. We demonstrate the proposed multifidelity approach by studying the mobility of a colloid in a micro-channel as a function of the colloid charge and of the size of the ions dissolved in the fluid.
We develop and demonstrate a new, hybrid simulation approach for charged fluids, which combines the accuracy of the nonlocal, classical density functional theory (cDFT) with the efficiency of the Poisson–Nernst–Planck (PNP) equations. The approach is motivated by the fact that the more accurate description of the physics in the cDFT model is required only near the charged surfaces, while away from these regions the PNP equations provide an acceptable representation of the ionic system. We formulate the hybrid approach in two stages. The first stage defines a coupled hybrid model in which the PNP and cDFT equations act independently on two overlapping domains, subject to suitable interface coupling conditions. At the second stage we apply the principles of the alternating Schwarz method to the hybrid model by using the interface conditions to define the appropriate boundary conditions and volume constraints exchanged between the PNP and the cDFT subdomains. Numerical examples with two representative examples of ionic systems demonstrate the numerical properties of the method and its potential to reduce the computational cost of a full cDFT calculation, while retaining the accuracy of the latter near the charged surfaces.
We present a new meshless method for scalar diffusion equations, which is motivated by their compatible discretizations on primal-dual grids. Unlike the latter though, our approach is truly meshless because it only requires the graph of nearby neighbor connectivity of the discretization points xi. This graph defines a local primal-dual grid complex with a virtual dual grid, in the sense that specification of the dual metric attributes is implicit in the method's construction. Our method combines a topological gradient operator on the local primal grid with a generalized moving least squares approximation of the divergence on the local dual grid. We show that the resulting approximation of the div-grad operator maintains polynomial reproduction to arbitrary orders and yields a meshless method, which attains O(hm) convergence in both L2- and H1-norms, similar to mixed finite element methods. We demonstrate this convergence on curvilinear domains using manufactured solutions in two and three dimensions. Application of the new method to problems with discontinuous coefficients reveals solutions that are qualitatively similar to those of compatible mesh-based discretizations.
We develop and analyze an optimization-based method for the coupling of nonlocal and local diffusion problems with mixed volume constraints and boundary conditions. The approach formulates the coupling as a control problem where the states are the solutions of the nonlocal and local equations, the objective is to minimize their mismatch on the overlap of the nonlocal and local domains, and the controls are virtual volume constraints and boundary conditions. When some assumptions on the kernel functions hold, we prove that the resulting optimization problem is well-posed and discuss its implementation using Sandia’s agile software components toolkit. As a result, the latter provides the groundwork for the development of engineering analysis tools, while numerical results for nonlocal diffusion in three-dimensions illustrate key properties of the optimization-based coupling method.
Formulation of locally conservative least-squares finite element methods (LSFEMs) for the Stokes equations with the no-slip boundary condition has been a long standing problem. Existing LSFEMs that yield exactly divergence free velocities require non-standard boundary conditions (Bochev and Gunzburger, 2009 [3]), while methods that admit the no-slip condition satisfy the incompressibility equation only approximately (Bochev and Gunzburger, 2009 [4, Chapter 7]). Here we address this problem by proving a new non-standard stability bound for the velocity-vorticity-pressure Stokes system augmented with a no-slip boundary condition. This bound gives rise to a norm-equivalent least-squares functional in which the velocity can be approximated by div-conforming finite element spaces, thereby enabling a locally-conservative approximations of this variable. We also provide a practical realization of the new LSFEM using high-order spectral mimetic finite element spaces (Kreeft et al., 2011) and report several numerical tests, which confirm its mimetic properties.
We present an abstract mathematical framework for an optimization-based additive decomposition of a large class of variational problems into a collection of concurrent subproblems. The framework replaces a given monolithic problem by an equivalent constrained optimization formulation in which the subproblems define the optimization constraints and the objective is to minimize the mismatch between their solutions. The significance of this reformulation stems from the fact that one can solve the resulting optimality system by an iterative process involving only solutions of the subproblems. Consequently, assuming that stable numerical methods and efficient solvers are available for every subproblem, our reformulation leads to robust and efficient numerical algorithms for a given monolithic problem by breaking it into subproblems that can be handled more easily. An application of the framework to the Oseen equations illustrates its potential.
We develop and analyze an optimization-based method for the coupling of nonlocal and local diffusion problems with mixed volume constraints and boundary conditions. The approach formulates the coupling as a control problem where the states are the solutions of the nonlocal and local equations, the objective is to minimize their mismatch on the overlap of the nonlocal and local domains, and the controls are virtual volume constraints and boundary conditions. When some assumptions on the kernel functions hold, we prove that the resulting optimization problem is well-posed and discuss its implementation using Sandia's agile software components toolkit. The latter provides the groundwork for the development of engineering analysis tools, while numerical results for nonlocal diffusion in three-dimensions illustrate key properties of the optimization-based coupling method.
ESAIM: Mathematical Modelling and Numerical Analysis
Olson, Derek; Shapeev, Alexander V.; Bochev, Pavel B.; Luskin, Mitchell
We formulate and analyze an optimization-based Atomistic-to-Continuum (AtC) coupling method for problems with point defects. Application of a potential-based atomistic model near the defect core enables accurate simulation of the defect. Away from the core, where site energies become nearly independent of the lattice position, the method switches to a more efficient continuum model. The two models are merged by minimizing the mismatch of their states on an overlap region, subject to the atomistic and continuum force balance equations acting independently in their domains. We prove that the optimization problem is well-posed and establish error estimates.