Large-scale Three-dimensional Problems Research Articles

Frequency-domain electromagnetic induction for the prediction of electrical conductivity and magnetic susceptibility using geostatistical inversion and randomized tensor decomposition

High-resolution characterization of near-surface systems is crucial for a variety of subsurface applications. Frequency-domain electromagnetic (FDEM) induction has been widely used in near-surface characterization compared with other geophysical methods due to its flexibility in acquisition and its ability to survey large areas with high resolution but with relatively low costs. FDEM measurements are sensitive to subsurface electrical conductivity (EC) and magnetic susceptibility (MS). However, the prediction of these properties requires solving a geophysical inverse problem. We combine ensemble smoother with multiple data assimilation (ES-MDA) and model reparameterization via randomized tensor decomposition (RTD) to simultaneously predict EC and MS from measured FDEM data. ES-MDA is an iterative data assimilation method that can be applied to nonlinear forward operators and provides multiple posterior realizations conditioned on the geophysical measurements to evaluate the model uncertainty. However, its application is usually computationally prohibitive for large-scale 3D problems. To overcome this limitation, we reduce the model parameters using RTD and then perform the inversion in the low-dimensional model space. The method is applied to synthetic and noisy real data sets. In the synthetic application example, the predicted posterior realizations illustrate the ability of our method to recover the true models of EC and MS accurately. The real case application comprises FDEM data acquired over arable land characterized by quaternary siliciclastic deposits with geoarchaeological features. We assess the performance of the inversion method at a borehole location not used to constrain the inversion. The inverted models do capture the available log data, illustrating the applicability of the inversion method to noisy real data.

A Problem-Independent Machine Learning (PIML) enhanced substructure-based approach for large-scale structural analysis and topology optimization of linear elastic structures

Structural analysis for structures composed of highly heterogeneous materials which often involves the solution of large-scale linear algebraic equations is very time-consuming even within the linear elastic regime. Furthermore, the tremendous computational cost of iterative large-scale finite element analysis also prevents the widespread use of topology optimization as a powerful design tool especially when the desired design resolution is very high. In order to break the bottleneck hindering the efficient solutions of large-scale structural analysis and design optimization problems, in the present article, a general machine learning (ML) enhanced substructure-based framework is proposed. The essential idea is resorting to the classical substructure-based finite element analysis approach and establishing an implicit mapping between the parameters characterizing the material distribution within a substructure and the corresponding condensed stiffness matrix/numerical shape functions through offline trained deep neural networks. In contrast to most of the existing ML enhanced approaches, the proposed framework is truly independent of the forms of structural geometry, boundary condition and external load, and can be applied to solve various boundary value problems governing by the same type of partial differential equation once the offline training is completed. Armed with modern artificial intelligence techniques, the proposed approach in some sense revives the classical substructure approach in finite element analysis. Compared with the traditional paradigm, it can achieve 104–105 times solution efficiency for tested large-scale examples with satisfactory accuracy. The effectiveness of the approach was also validated for the non-adjoint topology optimization problems, i.e., 3D compliant mechanism design. Finally, to demonstrate the proposed approach’s capability in dealing with extremely large-scale three-dimensional problems, a three-dimensional topology optimization problem with about 109 design variables and 3×109 degrees of freedom (Dofs) is solved on a laptop without resorting to any parallel computing techniques.

Parallel optimization of underwater acoustic models: A survey

Underwater acoustic models are effective tools for simulating underwater sound propagation. More than 50 years of research have been conducted on the theory and computational models of sound propagation in the ocean. Unfortunately, underwater sound propagation models were unable to solve practical large-scale three-dimensional problems for many years due to limited computing power and hardware conditions. Since the mid-1980s, research on high performance computing for acoustic propagation models in the field of underwater acoustics has flourished with the emergence of high-performance computing platforms, enabling underwater acoustic propagation models to solve many practical application problems that could not be solved before. In this paper, the contributions of research on high-performance computing for underwater acoustic propagation models since the 1980s are thoroughly reviewed and the possible development directions for the future are outlined.

Analysis of a Sinclair-Type Domain Decomposition Solver for Atomistic/Continuum Coupling

The "flexible boundary condition" method, introduced by Sinclair and coworkers in the 1970s, remains among the most popular methods for simulating isolated two-dimensional crystalline defects, embedded in an effectively infinite atomistic domain. In essence, the method can be characterized as a domain decomposition method which iterates between a local anharmonic and a global harmonic problem, where the latter is solved by means of the lattice Green function of the ideal crystal. This local/global splitting gives rise to tremendously improved convergence rates over related alternating Schwarz methods. In a previous publication (Hodapp et al., 2019, Comput. Methods in Appl. Mech. Eng. 348), we have shown that this method also applies to large-scale three-dimensional problems, possibly involving hundreds of thousands of atoms, using fast summation techniques exploiting the low-rank nature of the asymptotic lattice Green function. Here, we generalize the Sinclair method to bounded domains and develop an implementation using a discrete boundary element method to correct the infinite solution with respect to a prescribed far-field condition, thus preserving the advantage of the original method of not requiring a global spatial discretization. Moreover, we present a detailed convergence analysis and show for a one-dimensional problem that the method is unconditionally stable under physically motivated assumptions. To further improve the convergence behavior, we develop an acceleration technique based on a relaxation of the transmission conditions between the two subproblems. Numerical examples for linear and nonlinear problems are presented to validate the proposed methodology.

Augmented Lagrangian preconditioner for large-scale hydrodynamic stability analysis

Hydrodynamic linear stability analysis of large-scale three-dimensional configurations is usually performed with a“time-stepping” approach, based on the adaptation of existing solvers for the unsteady incompressible Navier–Stokes equations. We propose instead to solve the nonlinear steady equations with the Newton method and to determine the largest growth-rate eigenmodes of the linearized equations using a shift-and-invert spectral transformation and a Krylov–Schur algorithm. The solution of the shifted linearized Navier–Stokes problem, which is the bottleneck of this approach, is computed via an iterative Krylov subspace solver preconditioned by the modified augmented Lagrangian (mAL) preconditioner (Benzi et al., 2011). The well-known efficiency of this preconditioned iterative strategy for solving the real linearized steady-state equations is assessed here for the complex shifted linearized equations. The effect of various numerical and physical parameters is investigated numerically on a two-dimensional flow configuration, confirming the reduced number of iterations over state-of-the-art steady-state and time-stepping-based preconditioners. A parallel implementation of the steady Navier–Stokes and eigenvalue solvers, developed in the FreeFEM language, suitably interfaced with the PETSc/SLEPc libraries, is described and made openly available to tackle three-dimensional flow configurations. Its application on a small-scale three-dimensional problem shows the good performance of this iterative approach over a direct LU factorization strategy, in regards of memory and computational time. On a large-scale three-dimensional problem with 75 million unknowns, a 80% parallel efficiency on 256 up to 2048 processes is obtained.

A stable and high-order accurate discontinuous Galerkin based splitting method for the incompressible Navier–Stokes equations

In this paper we consider discontinuous Galerkin (DG) methods for the incompressible Navier–Stokes equations in the framework of projection methods. In particular we employ symmetric interior penalty DG methods within the second-order rotational incremental pressure correction scheme. The major focus of the paper is threefold: i) We propose a modified upwind scheme based on the Vijayasundaram numerical flux that has favourable properties in the context of DG. ii) We present a novel postprocessing technique in the Helmholtz projection step based on H(div) reconstruction of the pressure correction that is computed locally, is a projection in the discrete setting and ensures that the projected velocity satisfies the discrete continuity equation exactly. As a consequence it also provides local mass conservation of the projected velocity. iii) Numerical results demonstrate the properties of the scheme for different polynomial degrees applied to two-dimensional problems with known solution as well as large-scale three-dimensional problems. In particular we address second-order convergence in time of the splitting scheme as well as its long-time stability.

A high-performance parallel implementation of the certified reduced basis method

The certified reduced basis method (herein RB method) is a popular approach for model reduction of parametrized partial differential equations. In this paper we introduce new techniques that are required to efficiently implement the Offline “Construction stage” of the RB method on high-performance parallel supercomputers. This enables us to generate certified RB models for large-scale three-dimensional problems that can be evaluated on standard workstations and other “thin” computing resources with speedup of many orders of magnitude compared to the corresponding full order model. We use our implementation to perform detailed numerical studies for two computationally expensive model problems: a natural convection fluid flow problem and a “many parameter” heat transfer problem. In the heat transfer problem, we exploit the computational efficiency of the RB method to perform a detailed study of “snapshot” selection in the Greedy algorithm, and we also examine statistics of the output sensitivity derivatives to obtain a “global” view of the relative importance of the parameters.

A meshless method for conjugate heat transfer problems

Mesh reduction methods such as boundary element methods, method of fundamental solutions, and spectral methods all lead to fully populated matrices. This poses serious challenges for large-scale three-dimensional problems due to storage requirements and iterative solution of a large set of non-symmetric equations. Researchers have developed several approaches to address this issue including the class of fast-multipole techniques, use of wavelet transforms, and matrix decomposition. In this paper, we develop a domain decomposition, or the artificial sub-sectioning technique, along with a region-by-region iteration algorithm particularly tailored for parallel computation to address the coefficient matrix issue. The meshless method we employ is based on expansions using radial-basis functions (RBFs). An efficient physically based procedure provides an effective initial guess of the temperatures along the sub-domain interfaces. The iteration process converges very efficiently, offers substantial savings in memory, and features superior computational efficiency. The meshless iterative domain decomposition technique is ideally suited for parallel computation. We discuss its implementation under MPI standards on a small Windows XP PC cluster. Numerical results reveal the domain decomposition meshless methods produce accurate temperature predictions while requiring a much-reduced effort in problem preparation in comparison to other traditional numerical methods.

PARALLEL DOMAIN DECOMPOSITION APPROACH FOR LARGE-SCALE THREE-DIMENSIONAL BOUNDARY-ELEMENT MODELS IN LINEAR AND NONLINEAR HEAT CONDUCTION

The boundary-element method (BEM) requires only a surface mesh to solve linear and nonlinear heat conduction problems, but the resulting matrix is fully populated. This poses serious challenges for large-scale three-dimensional problems due to storage requirements and iterative solution of a large set of nonsymmetric equations. In this article, we develop a domain decomposition, or artificial subsectioning technique, along with a region-by-region iteration algorithm particularly tailored for parallel computation to address these issues. A coarse-surface grid solution coupled with an efficient physically based procedure provides an effective initial guess for a fine-surface grid model. The process converges very efficiently, offering substantial savings in memory. The iterative domain decomposition technique is ideally suited for parallel computation. We discuss its implementation on a modest Windows XP Pentium P4 PC cluster running under MPI with MPI2 extensions. Results from three-dimensional BEM heat conduction models including models of upwards of 85,000 nodes arising form an intricate film-cooled vane. We demonstrate that the BEM can readily be applied to solve large-scale linear and nonlinear heat conduction problems and that such solutions can be readily undertaken on modest PC clusters.

Dispersion-relation-preserving FDTD algorithms for large-scale three-dimensional problems

We introduce dispersion-relation-preserving (DRP) algorithms to minimize the numerical dispersion error in large-scale three-dimensional (3D) finite-difference time-domain (FDTD) simulations. The dispersion error is first expanded in spherical harmonics in terms of the propagation angle and the leading order terms of the series are made equal to zero. Frequency-dependent FDTD coefficients are then obtained and subsequently expanded in a polynomial (Taylor) series in the frequency variable. An inverse Fourier transformation is used to allow for the incorporation of the new coefficients into the FDTD updates. Butterworth or Chebyshev filters are subsequently employed to fine-tune the FDTD coefficients for a given narrowband or broadband range of frequencies of interest. Numerical results are used to compare the proposed 3D DRP-FDTD schemes against traditional high-order FDTD schemes.

A large-scale three-dimensional finite element analysis of incompressible viscous fluid flow

An efficient finite element scheme for large-scale three-dimensional flow analysis is proposed. Focus of attention is placed on the time integration algorithm and some techniques for the reduction of the storage requirement, including the one-point quadrature technique and an iterative matrix solver. Application to large-scale three-dimensional problems is given to demonstrate the feasibility of the proposed scheme.