Publications

Bochev, Pavel B.; Ridzal, Denis R.

Abstract not provided.

Siefert, Christopher S.; Kramer, Richard M.; Voth, Thomas E.; Bochev, Pavel B.

Abstract not provided.

Ridzal, Denis R.; Bochev, Pavel B.

Abstract not provided.

Proposed for publication in Applied Numerical Mathematics.

Bochev, Pavel B.; Dohrmann, Clark R.

Abstract not provided.

Bochev, Pavel B.; Moe, Scott M.; Peterson, Kara J.; Ridzal, Denis R.

Abstract not provided.

Journal of Computational Physics

Trask, Nathaniel A.; Bochev, Pavel B.; Perego, Mauro P.

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.

COUPLED PROBLEMS 2015 - Proceedings of the 6th International Conference on Coupled Problems in Science and Engineering

Bochev, Pavel B.; Moe, Scott A.; Peterson, Kara J.; Ridzal, Denis R.

We present a new optimization-based, conservative, and quasi-monotone method for passive tracer transport. The scheme combines high-order spectral element discretization in space with semi-Lagrangian time stepping. Solution of a singly linearly constrained quadratic program with simple bounds enforces conservation and physically motivated solution bounds. The scheme can handle efficiently a large number of passive tracers because the semi-Lagrangian time stepping only needs to evolve the grid points where the primitive variables are stored and allows for larger time steps than a conventional explicit spectral element method. Numerical examples show that the use of optimization to enforce physical properties does not affect significantly the spectral accuracy for smooth solutions. Performance studies reveal the benefits of high-order approximations, including for discontinuous solutions.

D'Elia, Marta D.; Perego, Mauro P.; Bochev, Pavel B.; Littlewood, David J.

Abstract not provided.

Perego, Mauro P.; D'Elia, Marta D.; Bochev, Pavel B.; Littlewood, David J.

Abstract not provided.

D'Elia, Marta D.; Littlewood, David J.; Perego, Mauro P.; Bochev, Pavel B.

Abstract not provided.

Computers and Mathematics with Applications (Oxford)

D'Elia, Marta D.; Perego, Mauro P.; Bochev, Pavel B.; Littlewood, David J.

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.

Computers and Mathematics with Applications

D'Elia, Marta D.; Perego, Mauro P.; Bochev, Pavel B.; Littlewood, David J.

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.

SIAM Journal on Scientific Computing

Trask, Nathaniel; Perego, Mauro P.; Bochev, Pavel B.

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.

Bochev, Pavel B.

Abstract not provided.

Journal of Computational Physics

Cheung, James C.; Frischknecht, Amalie F.; Perego, Mauro P.; Bochev, Pavel B.

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.

Lecture Notes in Computational Science and Engineering

Gerritsma, Marc; Bochev, Pavel B.

We present a locally conservative spectral least-squares formulation for the scalar diffusion-reaction equation in curvilinear coordinates. Careful selection of a least squares functional and compatible finite dimensional subspaces for the solution space yields the conservation properties. Numerical examples confirm the theoretical properties of the method.

International Journal for Numerical Methods in Fluids

Bochev, Pavel B.; Lai, James; Olson, Luke

Conventional least-squares finite element methods (LSFEMs) for incompressible flows conserve mass only approximately. For some problems, mass loss levels are large and result in unphysical solutions. In this paper we formulate a new, locally conservative LSFEM for the Stokes equations wherein a discrete velocity field is computed that is point-wise divergence free on each element. The central idea is to allow discontinuous velocity approximations and then to define the velocity field on each element using a local stream-function. The effect of the new LSFEM approach on improved local and global mass conservation is compared with a conventional LSFEM for the Stokes equations employing standard C 0 Lagrangian elements. © 2011 John Wiley & Sons, Ltd.

Bochev, Pavel B.; Christon, Mark A.; Collis, Samuel S.; Lehoucq, Richard B.; Shadid, John N.; Slepoy, Alexander S.

Existing approaches in multiscale science and engineering have evolved from a range of ideas and solutions that are reflective of their original problem domains. As a result, research in multiscale science has followed widely diverse and disjoint paths, which presents a barrier to cross pollination of ideas and application of methods outside their application domains. The status of the research environment calls for an abstract mathematical framework that can provide a common language to formulate and analyze multiscale problems across a range of scientific and engineering disciplines. In such a framework, critical common issues arising in multiscale problems can be identified, explored and characterized in an abstract setting. This type of overarching approach would allow categorization and clarification of existing models and approximations in a landscape of seemingly disjoint, mutually exclusive and ad hoc methods. More importantly, such an approach can provide context for both the development of new techniques and their critical examination. As with any new mathematical framework, it is necessary to demonstrate its viability on problems of practical importance. At Sandia, lab-centric, prototype application problems in fluid mechanics, reacting flows, magnetohydrodynamics (MHD), shock hydrodynamics and materials science span an important subset of DOE Office of Science applications and form an ideal proving ground for new approaches in multiscale science.

Bochev, Pavel B.; Collis, Samuel S.; Jones, Reese E.; Lehoucq, Richard B.; Parks, Michael L.; Scovazzi, Guglielmo S.; Silling, Stewart A.; Templeton, Jeremy A.

This report is a collection of documents written as part of the Laboratory Directed Research and Development (LDRD) project A Mathematical Framework for Multiscale Science and Engineering: The Variational Multiscale Method and Interscale Transfer Operators. We present developments in two categories of multiscale mathematics and analysis. The first, continuum-to-continuum (CtC) multiscale, includes problems that allow application of the same continuum model at all scales with the primary barrier to simulation being computing resources. The second, atomistic-to-continuum (AtC) multiscale, represents applications where detailed physics at the atomistic or molecular level must be simulated to resolve the small scales, but the effect on and coupling to the continuum level is frequently unclear.

Cheung, James C.; Frischknecht, Amalie F.; Perego, Mauro P.; Bochev, Pavel B.

Abstract not provided.

Computer Methods in Applied Mechanics and Engineering

Hughes, Thomas J.R.; Scovazzi, Guglielmo S.; Bochev, Pavel B.; Buffa, Annalisa

Proliferation of degrees-of-freedom has plagued discontinuous Galerkin methodology from its inception over 30 years ago. This paper develops a new computational formulation that combines the advantages of discontinuous Galerkin methods with the data structure of their continuous Galerkin counterparts. The new method uses local, element-wise problems to project a continuous finite element space into a given discontinuous space, and then applies a discontinuous Galerkin formulation. The projection leads to parameterization of the discontinuous degrees-of-freedom by their continuous counterparts and has a variational multiscale interpretation. This significantly reduces the computational burden and, at the same time, little or no degradation of the solution occurs. In fact, the new method produces improved solutions compared with the traditional discontinuous Galerkin method in some situations. © 2005 Elsevier B.V. All rights reserved.

Scovazzi, Guglielmo S.; Bochev, Pavel B.

Abstract not provided.

Scovazzi, Guglielmo S.; Bochev, Pavel B.

Proliferation of degrees-of-freedom has plagued discontinuous Galerkin methodology from its inception over 30 years ago. This paper develops a new computational formulation that combines the advantages of discontinuous Galerkin methods with the data structure of their continuous Galerkin counterparts. The new method uses local, element-wise problems to project a continuous finite element space into a given discontinuous space, and then applies a discontinuous Galerkin formulation. The projection leads to parameterization of the discontinuous degrees-of-freedom by their continuous counterparts and has a variational multiscale interpretation. This significantly reduces the computational burden and, at the same time, little or no degradation of the solution occurs. In fact, the new method produces improved solutions compared with the traditional discontinuous Galerkin method in some situations.

Bochev, Pavel B.; Perego, Mauro P.; Gao, Xujiao G.; Peterson, Kara J.

Abstract not provided.

Bochev, Pavel B.; Kuberry, Paul A.

Abstract not provided.