In this paper, we consider Maxwell’s equations in linear dispersive media described by a
single-pole Lorentz model for electronic polarization. We study two classes of commonly
used spatial discretizations: finite difference methods (FD) with arbitrary even order
accuracy in space and high spatial order discontinuous Galerkin (DG) finite element
methods. Both types of spatial discretizations are coupled with second order semi-implicit
leap-frog and implicit trapezoidal temporal schemes. By performing detailed dispersion
analysis for the semi-discrete and fully discrete schemes, we obtain rigorous quantification
of the dispersion error for Lorentz dispersive dielectrics. In particular, comparisons of
dispersion error can be made taking into account the model parameters, and mesh sizes
in the design of the two types of schemes. This work is a continuation of our previous
research on energy-stable numerical schemes for nonlinear dispersive optical media [6,7].
The results for the numerical dispersion analysis of the reduced linear model, considered
in the present paper, can guide us in the optimal choice of discretization parameters
for the more complicated and nonlinear models. The numerical dispersion analysis of
the fully discrete FD and DG schemes, for the dispersive Maxwell model considered
in this paper, clearly indicate the dependence of the numerical dispersion errors on
spatial and temporal discretizations, their order of accuracy, mesh discretization parameters
and model parameters. The results obtained here cannot be arrived at by considering
discretizations of Maxwell’s equations in free space. In particular, our results contrast the
advantages and disadvantages of using high order FD or DG schemes and leap-frog or
trapezoidal time integrators over different frequency ranges using a variety of measures
more »
« less
This content will become publicly available on May 30, 2024
Implementation and verification of a user‐defined element (UEL) for coupled thermal‐hydraulic‐mechanical‐chemical (THMC) processes in saturated geological media
Efficient and accurate modeling of the coupled thermal‐hydraulic‐mechanical‐chemical (THMC) processes in various rock formations is indispensable for designing energy geo‐structures such as underground repositories for high‐level nuclear wastes. This work focuses on developing and verifying an implicit finite element solver for generic coupled THMC problems in geological settings. Starting from the mass, momentum, and energy balance laws, a specialized set of governing equations and a thermoporoelastic constitutive model is derived. This system is then solved by an implicit finite element (FE) scheme. Specifically, the residuals and the Jacobians are scripted in a user‐defined element (UEL) subroutine which is then combined with the general‐purpose FE software Abaqus Standard to solve initial‐boundary value problems. Considering the complexity of the system, the UEL development follows a stepwise manner by first solving the coupled hydraulic‐mechanical (HM) and thermal‐hydraulic‐mechanical (THM) equations before moving on to the full THMC problem. Each implementation step consists of at least one verification test by comparing computed results with closed‐form analytical solutions to ensure that the various coupling effects are correctly realized. To demonstrate the robustness of the algorithm and to validate the UEL, a three‐dimensional case study is performed with reference to the in‐situ heating test of ATLAS at Belgium in 1980s. A hypothetical radionuclide leakage event is then simulated by activating the chemical‐concentration degree of freedom and prescribing a constant high concentration at the heater's surface. The model predicts a limited contaminated regime after six years considering both diffusion and advection effects on species transport.
more » « less- Award ID(s):
- 2113474
- NSF-PAR ID:
- 10419296
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- International Journal for Numerical and Analytical Methods in Geomechanics
- Volume:
- 47
- Issue:
- 11
- ISSN:
- 0363-9061
- Page Range / eLocation ID:
- p. 2153-2190
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
more » « less
Abstract State‐of‐the‐art distributed‐memory computer clusters contain multicore CPUs with 16 and more cores. The second generation of the Intel Xeon Phi many‐core processor has more than 60 cores with 16 GB of high‐performance on‐chip memory. We contrast the performance of the second‐generation Intel Xeon Phi, code‐named Knights Landing (KNL), with 68 computational cores to the latest multicore CPU Intel Skylake with 18 cores. A special‐purpose code solving a system of nonlinear reaction‐diffusion partial differential equations with several thousands of point sources modeled mathematically by Dirac delta distributions serves as realistic test bed. The system is discretized in space by the finite volume method and advanced by fully implicit time‐stepping, with a matrix‐free implementation that allows the complex model to have an extremely small memory footprint. The sample application is a seven variable model of calcium‐induced calcium release (CICR) that models the interplay between electrical excitation, calcium signaling, and mechanical contraction in a heart cell. The results demonstrate that excellent parallel scalability is possible on both hardware platforms, but that modern multicore CPUs outperform the specialized many‐core Intel Xeon Phi KNL architecture for a large class of problems such as systems of parabolic partial differential equations.
-
more » « less
Abstract Computational fluid dynamics (CFD) is increasingly used to study blood flows in patient‐specific arteries for understanding certain cardiovascular diseases. The techniques work quite well for relatively simple problems but need improvements when the problems become harder when (a) the geometry becomes complex (eg, a few branches to a full pulmonary artery), (b) the model becomes more complex (eg, fluid‐only to coupled fluid‐structure interaction), (c) both the fluid and wall models become highly nonlinear, and (d) the computer on which we run the simulation is a supercomputer with tens of thousands of processor cores. To push the limit of CFD in all four fronts, in this paper, we develop and study a highly parallel algorithm for solving a monolithically coupled fluid‐structure system for the modeling of the interaction of the blood flow and the arterial wall. As a case study, we consider a patient‐specific, full size pulmonary artery obtained from computed tomography (CT) images, with an artificially added layer of wall with a fixed thickness. The fluid is modeled with a system of incompressible Navier‐Stokes equations, and the wall is modeled by a geometrically nonlinear elasticity equation. As far as we know, this is the first time the unsteady blood flow in a full pulmonary artery is simulated without assuming a rigid wall. The proposed numerical algorithm and software scale well beyond 10 000 processor cores on a supercomputer for solving the fluid‐structure interaction problem discretized with a stabilized finite element method in space and an implicit scheme in time involving hundreds of millions of unknowns.
-
Frost heave can cause serious damage to civil infrastructure. For example, interactions of soil and water pipes under frozen conditions have been found to significantly accelerate pipe fracture. Frost heave may cause the retaining walls along highways to crack and even fail in cold climates. This paper describes a holistic model to simulate the temperature, stress, and deformation in frozen soil and implement a model to simulate frost heave and stress on water pipelines. The frozen soil behaviors are based on a microstructure-based random finite element model, which holistically describes the mechanical behaviors of soils subjected to freezing conditions. The new model is able to simulate bulk behaviors by considering the microstructure of soils. The soil is phase coded and therefore the simulation model only needs the corresponding parameters of individual phases. This significantly simplifies obtaining the necessary parameters for the model. The capability of the model in simulating the temperature distribution and volume change are first validated with laboratory scale experiments. Coupled thermal-mechanical processes are introduced to describe the soil responses subjected to sub-zero temperature on the ground surface. This subsequently changes the interaction modes between ground and water pipes and leads to increase of stresses on the water pipes. The effects of cracks along a water pipe further cause stress concentration, which jeopardizes the pipe’s performance and leads to failure. The combined effects of freezing ground and traffic load are further evaluated with the model.more » « less
-
The Finite Element Method (FEM) is widely used to solve discrete Partial Differential Equations (PDEs) in engineering and graphics applications. The popularity of FEM led to the development of a large family of variants, most of which require a tetrahedral or hexahedral mesh to construct the basis. While the theoretical properties of FEM basis (such as convergence rate, stability, etc.) are well understood under specific assumptions on the mesh quality, their practical performance, influenced both by the choice of the basis construction and quality of mesh generation, have not been systematically documented for large collections of automatically meshed 3D geometries. We introduce a set of benchmark problems involving most commonly solved elliptic PDEs, starting from simple cases with an analytical solution, moving to commonly used test problem setups, and using manufactured solutions for thousands of real-world, automatically meshed geometries. For all these cases, we use state-of-the-art meshing tools to create both tetrahedral and hexahedral meshes, and compare the performance of different element types for common elliptic PDEs. The goal of this benchmark is to enable comparison of complete FEM pipelines, from mesh generation to algebraic solver, and exploration of relative impact of different factors on the overall system performance. As a specific application of our geometry and benchmark dataset, we explore the question of relative advantages of unstructured (triangular/ tetrahedral) and structured (quadrilateral/hexahedral) discretizations. We observe that for Lagrange-type elements, while linear tetrahedral elements perform poorly, quadratic tetrahedral elements perform equally well or outperform hexahedral elements for our set of problems and currently available mesh generation algorithms. This observation suggests that for common problems in structural analysis, thermal analysis, and low Reynolds number flows, high-quality results can be obtained with unstructured tetrahedral meshes, which can be created robustly and automatically. We release the description of the benchmark problems, meshes, and reference implementation of our testing infrastructure to enable statistically significant comparisons between different FE methods, which we hope will be helpful in the development of new meshing and FEA techniques.more » « less
