Fast estimation of discretization error for FE problems solved by domain decomposition

A non-overlapping Schwarz domain decomposition method with high-order finite elements for flow acoustics

A non-overlapping and non-conformal domain decomposition method (DDM) is presented for modelling large finite antenna arrays. There are two major ingredients in the proposed DDM: (a) A new second-order transmission condition is introduced, which improves convergence of the iterative process. In contrast to previous high order interface conditions, the new condition uses two second-order transverse derivatives to address the slow convergence issue of both TE and TM evanescent modes. Numerical experiments demonstrate that the convergence of the proposed algorithm is quite insensitive to the size of array. (b) The proposed non-conformal DDM not only permits the use of completely independent discretization for each of the sub-domains, but also allows adjacent sub-domains to be geometrically non-conformal. The benefits of the non-conformal nature of the proposed DDM will be fully enjoyed by a large-scale problem of practical interest, which is a 50 by 50 ultra wide band (UWB) array in the presence of a slot frequency selective surface (FSS). Numerical results verify the effectiveness of the proposed method.

A posteriori error analysis for two non-overlapping domain decomposition techniques

Numerical solution of neutral functional differential equations by Adams methods in divided difference form

A non-overlapping domain decomposition method with non-matching grids for modeling large finite antenna arrays

Domain decomposition based finite element verification in linear framework

Advances in Adaptive Computational Methods in Mechanics

Advances in interconnect technologies, such as the increase of the number of metal layers and 3-D stacking technique, have paved the way for higher functionality and superior performance while reducing size, power, and cost in today's integrated circuits and package products. With the increase of clock frequency and edge rates as well as the continuously downscaling of feature size and 3-D interconnect technologies in high-speed systems, signal integrity (SI) effects such as signal delay, reflection, attenuation, dispersion and crosstalk have become one of the dominant factors in current deep sub-micrometer CMOS technologies limiting overall performance of high-speed systems.Another major difficulty is the power integrity (PI) issue. For example, IR drop is caused by the finite resistivity (R) of metals and current (I) drawn off from the power/ground planes. Chip designs are susceptible to IR drop, especially when the integrated circuit (IC) supply voltage Vdd decreases with the scaling of silicon processes. IR drop is proportional to the resistance of the power/ground plane. When the resistance of power/ground plane increases, with the shrinking of complex geometries, the IR drops will in turn increase as well. To further confound the difficult issue, the rise of the thermal temperature due to the current carrying interconnects also has tremendous impact on the IC performance and reliability. Current flow in a VLSI interconnect can cause a power dissipation, which is referred to as Joule heating or self-heating. The Joule heating effects are becoming increasingly significant with the shrinking scale of silicon process because of the increase of on-chip power density, inclusion of more metal layers with higher densities, and the use of dielectric materials with lower thermal conductivities. Subsequently, the temperature effects on the electrical performance of 3D systems should be carefully considered as well in the electrical designs.Unfortunately, almost all practical SI and PI problems are multi-physics in nature, and various physical phenomena are usually interacting and coupling to each other. For instance, the resistivity of most conducting metals increases linearly with the increases of the surrounding temperature resulting from the Joule heating by electric currents flowing through the conductors. Therefore, in order to accurately characterize the performance of high power integrated circuits (ICs), packages and printed circuit boards (PCBs), it is essential to account for both electric and thermal effects and the intimate couplings between them.In this paper, we propose to use non-conformal, non-overlapping domain decomposition methods (DDMs) for the signal integrity analyses and thermal-aware DC IR drop co-analysis of high-power IC-Package-PCBs. The proposed DDM starts by partitioning the composite device into inhomogeneous sub-regions with temperature dependent material properties. Subsequently, each sub-domain is meshed independently according to its own characteristic features. As a result, the troublesome mesh generation task for complex ICs can be greatly subdued. Moreover, the proposed thermal-aware DC IR drop co-analysis applies the non-conformal DDMs for both conduction and steady state heat transfer analyses with a two-way coupling between them. Furthermore, we extend the non-conformal DDM to study transient thermal temperature distribution within complex ICs. Numerical examples, including an IC package and an IC-Package-PCB, demonstrate the flexibility and potentials of the proposed non-conformal DDMs for SI analyses and thermal-aware DC IR-drop co-analysis.

In this article, a nonconformal and nonoverlapping domain decomposition method (DDM) based on surface integral equation (SIE) is proposed for the electromagnetic (EM) scattering or radiation of multiscale metallic targets. To reduce the unknown amount of the conventional SIE for multiscale EM simulation, we apply curved triangular elements and higher order hierarchical vector (HOHV) basis functions to the SIE. Next, to increase the flexibility of geometrical modeling and accelerate the convergence of the presented SIE system for electrically large and multiscale metallic targets, a DDM scheme is further developed to employ a discontinuous Galerkin (DG) approach to glue conformal/nonconformal surface grids between adjacent subdomains. In addition, an interior penalty term is introduced to stabilize the DDM solution, and half edge-based HOHV basis functions are introduced to model the current associated with nonconformal surface elements. Meanwhile, the basis expansion and recombination (BER) technique is introduced to significantly accelerate the matrix-filling and improve the efficiency. The flexibility of basis order selection is further enhanced by the hierarchical characteristic of HOHV bases. Finally, several numerical results are provided to demonstrate the accuracy, efficiency, and flexibility of the proposed HO-DG-DDM.

Computational Fluid Dynamics (CFD) has gained significant utility in the analysis of diverse fluid flow scenarios, thanks to advances in computational power. Accurate error estimation techniques are crucial to ensure the reliability of CFD simulations, as errors can lead to misleading conclusions. This study focuses on the estimation of discretization errors in time-dependent simulations, building upon prior work addressing steady-state problems Wang et al. (2020, “Error Transport Equation Implementation in the Sensei CFD Code,” AIAA Paper No. 2020-1047). In this research, we employ unsteady error transport equations (ETE) to generate localized discretization error estimates within the framework of the finite volume CFD code SENSEI. For steady-state problems, the ETE only need to be solved once after the solution has converged, whereas the unsteady ETE need to be co-advanced with the primal solve. To enhance efficiency, we adopt a one-sided temporal stencil and develop a modified iterative correction process tailored to the unsteady ETE. The time-marching schemes utilized encompass second-order accurate singly diagonally implicit Runge–Kutta (SDIRK) and second-order backward differentiation formula (BDF2), both being implicit. To rigorously assess the accuracy of our error estimates, all test cases feature known analytical solutions, facilitating order-of-accuracy evaluations. Two test cases are considered: the 2D convected vortex for inviscid flow and a cross-term sinusoidal (CTS) manufactured solution for viscous flow. Results indicate higher-order convergence rates for the 2D convected vortex test case even when iterative correction is not applied, with similar observations in the CTS case, albeit not at the finest grid levels. Although the current implementation of iterative correction exhibits lower stability compared to the primal solve, it generally enhances the discretization error estimate. Notably, after iterative correction, the discretization error estimate for the unsteady ETE achieves higher-order accuracy across all grid levels in the 2D CTS manufactured solution.

In this paper, we are concerned with the non-overlapping domain decomposition method with non-matching grids for three-dimensional elliptic equations. For this method, the interface variable is chosen as the mortar multiplier. We first develop a class of inexact solvers for the interface Poincare-Steklov operator associated with such a domain decomposition. Then we use the inexact solver to construct a new preconditioner for the interface problem arising from the Finite Element Tearing and Interconnecting method. It will be shown that the condition number of the preconditioned system grows only as the logarithm of the dimension of the local problem associated with an individual substructure, and is independent of possible jumps of the coefficient in the elliptic equation.

In recent years, there has been gratifying progress in the development of domain decomposition algorithms for symmetric and nonsymmetric elliptic problems and even some indefinite problems. Many methods possess the attractive property that the convergence rate is optimal, i.e., independent of the size of the discrete problem and of the number of subdomains, or within a polylog factor of optimal. There is, in comparison, relatively little in the domain decomposition literature on hyperbolic problems. Quarteroni [8, 9] used nonoverlapping domain decomposition methods based on the spectral collocation approximation on systems of conservation laws. Gastaldi and Gastaldi [5, 6] set up a nonoverlapping domain decomposition scheme based on the finite element approximation for the transport equation. These contributions establish the boundary operators that lead to well-posed decoupled problems, which can then be discretized and solved by standard means. Our interests in this paper are rather different. We examine overlapping domain decomposition preconditioners, and leave the original global discretization fully in tact. Rather than deriving interface conditions that lead to decomposed solutions that are mathematically equivalent (to within some specified discretization tolerance) to the solutions of the undecomposed problem, we derive an approximate inverse that can be applied in a concurrent manner, subdomain-by-subdomain, and that effectively preconditions the original undecomposed operator, whose action is already trivial to apply in the same concurrent manner. There seem to have been to date no such additive or multiplicative Schwarz preconditioners leading to optimal convergence rates for hyperbolic equations. Based on the standard Galerkin method [4] an ASM algorithm is formulated. The preconditioned problems are solved by the GMRES method. The convergence

The Method of Nearby Problems (MNP) is employed to evaluate various discretization error estimators. Steady- state Burgers equation is chosen as the example case. Fifth-order Hermite splines are used to generate the analytic curve fit, and the development of these splines in 1D is discussed. Results are presented for Burgers equation corresponding to a viscous shock wave for Reynolds numbers of 8, 64, and 512, as well as for a modified version of Burgers equation with a variable viscosity at a nominal Reynolds number of 64. The results obtained using Hermite splines are compared with the results obtained using cubic splines as well as global Legendre polynomials. Whereas the Legendre polynomial fits exhibited large errors at the boundaries, Hermite splines provide good approximations over the entire domain. Furthermore, in contrast to cubic splines, Hermite splines are shown to provide slope- continuous source terms due to the use of C 3 continuity at spline zone interfaces. The fifth-order Hermite splines are used to generate an exact solution to a problem nearby the original Burgers equation. Various discretization error estimators are evaluated both for the original Burgers equation, the nearby problem, and a version of Burgers equation with a nonlinear viscosity term. The Method of Nearby Problems is also examined as an error estimator.

A non-overlapping domain decomposition method is applied to solve surface integral equations for electromagnetic scattering from cavities. This scheme allows the cavity to be divided into sub-domains and can be solved in a serial or parallel manner. Thus, the memory requirement and CPU time for modeling cavity structures can be dramatically reduced. Moreover, the interior resonance problem is eliminated from this scheme, which occurs in the generalized network formulation.

It is shown that for elliptic boundary value problems of order 2m the condition number of the Schur complement matrix that appears in nonoverlapping domain decomposition methods is of order \(d^{-1}h^{-2m+1}\), where d measures the diameters of the subdomains and h is the mesh size of the triangulation. The result holds for both conforming and nonconforming finite elements.