The independent set perturbation adjoint method: A new method of differentiating mesh‐based fluids models

A new scheme for differentiating complex mesh‐based numerical models (e.g. finite element models), the Independent Set Perturbation Adjoint method (ISP‐Adjoint), is presented. Differentiation of the matrices and source terms making up the discrete forward model is realized by a graph coloring approach (forming independent sets of variables) combined with a perturbation method to obtain gradients in numerical discretizations. This information is then convolved with the ‘mathematical adjoint’, which uses the transpose matrix of the discrete forward model. The adjoint code is simple to implement even with complex governing equations, discretization methods and non‐linear parameterizations. Importantly, the adjoint code is independent of the implementation of the forward code. This greatly reduces the effort required to implement the adjoint model and maintain it as the forward model continues to be developed; as compared with more traditional approaches such as applying automatic differentiation tools. The approach can be readily extended to reduced‐order models. The method is applied to a one‐dimensional Burgers' equation problem, with a highly non‐linear high‐resolution discretization method, and to a two‐dimensional, non‐linear, reduced‐order model of an idealized ocean gyre. Copyright © 2010 John Wiley & Sons, Ltd.     

Verification of Euler/Navier–Stokes codes using the method of manufactured solutions

The method of manufactured solutions is used to verify the order of accuracy of two finite‐volume Euler and Navier–Stokes codes. The Premo code employs a node‐centred approach using unstructured meshes, while the Wind code employs a similar scheme on structured meshes. Both codes use Roe's upwind method with MUSCL extrapolation for the convective terms and central differences for the diffusion terms, thus yielding a numerical scheme that is formally second‐order accurate. The method of manufactured solutions is employed to generate exact solutions to the governing Euler and Navier–Stokes equations in two dimensions along with additional source terms. These exact solutions are then used to accurately evaluate the discretization error in the numerical solutions. Through global discretization error analyses, the spatial order of accuracy is observed to be second order for both codes, thus giving a high degree of confidence that the two codes are free from coding mistakes in the options exercised. Examples of coding mistakes discovered using the method are also given. Copyright © 2004 John Wiley & Sons, Ltd.     

Laboratory astrophysics and non-ideal equations of state: the next challenges for astrophysical MHD simulations

Laboratory astrophysics holds great promise not only as a highly effective validation tool for astrophysical magneto-hydrodynamics (MHD) codes but it also presents a unique challenge for these codes. The high-density plasmas found in these experiments are not well modeled by the ideal equations of state (EOS) found in most astrophysical simulation codes. To solve this problem, we replaced the ideal EOS scheme in an existing MHD code, AstroBEAR, with a non-ideal EOS method and validated our implementation with van der Waals shock tube tests. The improved code is also able to model flows that contain more than one material, as required in laboratory experiments. Simulations of jet experiments performed at the OMEGA Laser reproduce the morphology of the jet much better than when the code used a single material and an ideal EOS.     

Critical evaluation of CFD codes for interfacial simulation of bubble‐train flow in a narrow channel

Computational fluid dynamics (CFD) codes that are able to describe in detail the dynamic evolution of the deformable interface in gas–liquid or liquid–liquid flows may be a valuable tool to explore the potential of multi‐fluid flow in narrow channels for process intensification. In the present paper, a computational exercise for co‐current bubble‐train flow in a square vertical mini‐channel is performed to investigate the performance of well‐known CFD codes for this type of flows. The computations are based on the volume‐of‐fluid method (VOF) where the transport equation for the liquid volumetric fraction is solved either by the methods involving a geometrical reconstruction of the interface or by the methods that use higher‐order difference schemes instead. The codes contributing to the present code‐to‐code comparison are an in‐house code and the commercial CFD packages CFX, FLUENT and STAR‐CD. Results are presented for two basic cases. In the first one, the flow is driven by buoyancy only, while in the second case the flow is additionally forced by an external pressure gradient. The results of the code‐to‐code comparison show that only the VOF method with interface reconstruction leads to physically sound and consistent results, whereas the use of difference schemes for the volume fraction equation shows some deficiencies. Copyright © 2007 John Wiley & Sons, Ltd.     

Improvement,validation and application of CFD/DEM model to dense gas–solid flow in a fluidized bed

Dense gas–solid flow with solid volume fraction greater than 10% and at moderate Reynolds number is important in many industrial facilities such as fluidized beds. In this work, the Euler–Lagrange approach in combination with a deterministic collision model is applied to a laboratory-scale fluidized bed. The fluid–particle interaction is studied using a new procedure called the offset method, which results in several numbers of spatial displacements of the fluid grid. The proposed method is highly precise in determining porosity and momentum transfer, thus improving simulation accuracy. A validation study was carried out to assess the results using this in-house CFD/DEM code against 5-s operation of a Plexiglas spouted-fluidized bed, showing good qualitative correlation of solid distribution in the bed and acceptable quantitative agreement of pressure drops at different positions in the bed. In view of high computing cost, special emphasis is placed on effective program design, such as application of advanced detection algorithm for particle–particle/wall collisions, the multi-grid method and parallel calculation. In this context, the influence of increasing the processor number, up to 36, on calculation efficiency was investigated.     

Semi‐automatic porting of a large‐scale CFD code to multi‐graphics processing unit clusters

A typical large‐scale CFD code based on adaptive, edge‐based finite‐element formulations for the solution of compressible and incompressible flow is taken as a test bed to port such codes to graphics hardware (graphics processing units, GPUs) using semi‐automatic techniques. In previous work, a GPU version of this code was presented, in which, for many run configurations, all mesh‐sized loops required throughout time stepping were ported. This approach simultaneously achieves the fine‐grained parallelism required to fully exploit the capabilities of many‐core GPUs, completely avoids the crippling bottleneck of GPU–CPU data transfer, and uses a transposed memory layout to meet the distinct memory access requirements posed by GPUs. The present work describes the next step of this porting effort, namely to integrate GPU‐based, fine‐grained parallelism with Message‐Passing‐Interface‐based, coarse‐grained parallelism, in order to achieve a code capable of running on multi‐GPU clusters. This is carried out in a semi‐automated fashion: the existing Fortran–Message Passing Interface code is preserved, with the translator inserting data transfer calls as required. Performance benchmarks indicate up to a factor of 2 performance advantage of the NVIDIA Tesla M2050 GPU (Santa Clara, CA, USA) over the six‐core Intel Xeon X5670 CPU (Santa Clara, CA, USA), for certain run configurations. In addition, good scalability is observed when running across multiple GPUs. The approach should be of general interest, as how best to run on GPUs is being presently considered for many so‐called legacy codes. Copyright © 2011 John Wiley & Sons, Ltd.     

A two‐step monolithic method for the efficient simulation of incompressible flows

We propose a simple technique for improving computationally the efficiency of monolithic velocity–pressure solvers for incompressible flow problems. The idea consists in solving the discrete nonlinear system of governing equations in two steps: introducing ‘artificial’ compressibility first and afterwards correcting the solution by solving the original incompressible system. The speed‐up is obtained because of a better conditioning of the modified discrete system solved at the prediction step. The formulation can be easily implemented into existing monolithic codes requiring minor modification only. The paper concludes with two examples validating the formulation and facilitating the estimation of the obtained speed‐up. For the tests chosen, an average speed‐up is approximately double, suggesting that the method is a feasible approach for incompressible flows' simulation. Copyright © 2014 John Wiley & Sons, Ltd.     

Parallel solution of a coupled flow and transport model for shallow water

We present an integrated approach for the concurrent solution of a 3D hydrodynamical model coupled with a 3D transport model. Since both models are quite similar in nature, the same numerical method has been employed. This leads to a code that is more efficient than when two existing codes would have been combined. Discretization of the spatial differential operators, and the boundary conditions, results in a stiff initial value problem. To cope with the stiffness, we select an implicit time‐integration formula, viz. the second‐order, L‐stable BDF method because of its excellent stability properties. To reduce the huge amount of linear algebra involved in solving the implicit relations, an Approximate Factorization technique has been used. Essentially, this technique replaces a ‘multi‐dimensional’ system by a series of ‘one‐dimensional’ systems. Since the output of the hydrodynamical model (i.e., the flow field) serves as input for the transport model, we solve the hydrodynamical model one time step ahead in time. This allows us to solve the models in parallel, using two different groups of processors. By a little tuning of the parameters in the algorithm, a load‐balancing has been obtained that is close to optimal. As a result, both models require roughly the same amount of CPU time, so that one of them, effectively, can be considered as obtained ‘for free’. Copyright © 2002 John Wiley & Sons, Ltd.     

基于知识图谱和GPT模型的可靠性代码自动生成方法

工程结构服役中广泛使用可靠性分析进行结构安全评估,但可靠性分析方法种类多、分析程序代码自动化程度低且复用难,需要研究可靠性代码自动生成方法。生成式预训练转换器GPT(Generative Pre-trained Transformer)模型已经在大量替代编程手工作业,进行代码自动生成。但是,其在工程领域中的应用受限于可学习数据量小和问题匹配度不高。本文提出了一种结合多种类可靠性知识图谱、基于GPT的代码自动完成模型进行Matlab可靠性代码预测的方法,使用精心设计的源代码预处理降噪策略,以及知识图谱传播模拟密集型任务解释意图;采用条件代码生成训练,有效提升了小数据样本量的学习性能,实现高准确率、问题匹配的可靠性代码自动生成。最后通过三个可靠性知识图谱案例验证了所提方法的有效性。     

Application of strain rate potentials with multiple linear transformations to the description of polycrystal plasticity

This paper reviews a class of anisotropic plastic strain-rate potentials, based on linear transformations of the plastic strain-rate tensor. A new formulation is proposed, which includes former models as particular cases and allows for an arbitrary number of linear transformations, involving an increasing number of anisotropy parameters. The formulation is convex and fully three-dimensional, thus being suitable for computer implementation in finite element codes. The parameter identification procedure uses a micromechanical model to generate evenly distributed reference points in the full space of possible loading modes. Material parameters are determined for several anisotropic, fcc and bcc sheet metals, and the gain in accuracy of the new models is demonstrated. For the considered materials, increasing the number of linear transformations leads to a systematic improvement of the accuracy, up to a number of five linear transformations. The proposed model fits very closely the predictions of the micromechanical model in the whole space of plastic strain-rate directions. The r-values, which are not directly used in the identification procedure, served for the validation of the models and to demonstrate their improved accuracy.     

Tools for simulating non-stationary incompressible flow via discretely divergence-free finite element models

We develop simulation tools for the non-stationary incompressible 2D Navier--Stokes equations. The most important components of the finite element code are: the fractional step ?-scheme, which is of second-order accuracy and strongly A-stable, for the time discretization; a fixed point defect correction method with adaptive step length control for the non-linear problems (stationary Navier-Stokes equations); a modified upwind discretization of higher-order accuracy for the convective terms. Finally, the resulting nonsymmetric linear subproblems are treated by a special multigrid algorithm which is adapted to the quadrilateral non-conforming discretely divergence-free finite elements. For the graphical postprocess we use a fully non-stationary and interactive particle-tracing method. With extensive test calculations we show that our method is a candidate for a ‘black box’ solver.     

Validation of numerical solvers for liquid metal flow in a complex geometry in the presence of a strong magnetic field

Following the magnetohydrodynamic (MHD) code validation and verification proposal by Smolentsev et al. (Fusion Eng Des 100:65–72, 2015), we perform code to code and code to experiment comparisons between two computational solvers, FLUIDYN and HIMAG, which are presently considered as two of the prospective CFD tools for fusion blanket applications. In such applications, an electrically conducting breeder/coolant circulates in the blanket ducts in the presence of a strong plasma-confining magnetic field at high Hartmann numbers, \(\textit{Ha}\) (\(\textit{Ha}^2\) is the ratio between electromagnetic and viscous forces) and high interaction parameters, \(\textit{N}\) (\(\textit{N}\) is the ratio of electromagnetic to inertial forces). The main objective of this paper is to provide the scientific and engineering community with common references to assist fusion researchers in the selection of adequate computational means to be used for blanket design and analysis. As an initial validation case, the two codes are applied to the classic problem of a laminar fully developed MHD flows in a rectangular duct. Both codes demonstrate a very good agreement with the analytical solution for \(\textit{Ha}\) up to 15, 000. To address the capabilities of the two codes to properly resolve complex geometry flows, we consider a case of three-dimensional developing MHD flow in a geometry comprising of a series of interconnected electrically conducting rectangular ducts. The computed electric potential distributions for two flows (Case A) \(\textit{Ha}=515\), \(\textit{N}=3.2\) and (Case B) \(\textit{Ha}=2059\), \(\textit{N}=63.8\) are in very good agreement with the experimental data, while the comparisons for the MHD pressure drop are still unsatisfactory. To better interpret the observed differences, the obtained numerical data are analyzed against earlier theoretical and experimental studies for flows that involve changes in the relative orientation between the flow and the magnetic field.     

Portable implementation model for CFD simulations. Application to hybrid CPU/GPU supercomputers

Nowadays, high performance computing (HPC) systems experience a disruptive moment with a variety of novel architectures and frameworks, without any clarity of which one is going to prevail. In this context, the portability of codes across different architectures is of major importance. This paper presents a portable implementation model based on an algebraic operational approach for direct numerical simulation (DNS) and large eddy simulation (LES) of incompressible turbulent flows using unstructured hybrid meshes. The strategy proposed consists in representing the whole time-integration algorithm using only three basic algebraic operations: sparse matrix–vector product, a linear combination of vectors and dot product. The main idea is based on decomposing the nonlinear operators into a concatenation of two SpMV operations. This provides high modularity and portability. An exhaustive analysis of the proposed implementation for hybrid CPU/GPU supercomputers has been conducted with tests using up to 128 GPUs. The main objective consists in understanding the challenges of implementing CFD codes on new architectures.     

Light element opacities from ATOMIC

We present new calculations of local-thermodynamic-equilibrium (LTE) light element opacities from the Los Alamos ATOMIC code. ATOMIC is a multi-purpose code that can generate LTE or non-LTE quantities of interest at various levels of approximation. A program of work is currently underway to compute new LTE opacity data for all elements H through Zn. New opacity tables for H through Ne are complete, and a new Fe opacity table will be available soon. Our calculations, which include fine-structure detail, represent a systematic improvement over previous Los Alamos opacity calculations using the LEDCOP legacy code. Our opacity calculations incorporate atomic structure data computed from the CATS code, which is based on Cowan's atomic structure codes, and photoionization cross section data computed from the Los Alamos ionization code GIPPER. We make use of a new equation-of-state (EOS) model based on the chemical picture. ATOMIC incorporates some physics packages from LEDCOP and also includes additional physical processes, such as improved free–free cross sections and additional scattering mechanisms. In this report, we briefly discuss the physics improvements included in our new opacity calculations and present comparisons of our new opacities with other work for C, O, and Fe at selected conditions.     

Active Slip System Identification in Polycrystalline Metals by Digital Image Correlation (DIC)

In this paper, a novel approach was proposed to increase the confidence of active slip system identification in polycrystalline metals. The approach takes advantage of microscale deformation tracking via Digital Image Correlation (DIC) combined with scanning electron microscopy (SEM). The experimentally-obtained high-resolution deformation fields were mapped to an undeformed configuration, which gives slip traces suitable for comparison with undeformed crystal orientation data. A metric, named herein as the ‘relative displacement ratio’ (RDR), is calculated from the displacement fields near slip traces to characterize the localized deformation due to slip. In validation cases, the experimentally-measured RDRs matched well with RDRs theoretically-calculated from active slip systems. In test cases, active slip system identification by incorporating RDR as an additional constraint was demonstrated to be preferable to using Schmid factor alone as a constraint. The proposed approach supplements existing techniques for slip system identification with increased confidence.     

A semi-collocated ocean model based on the SOMS approach

In application to the Gulf of Mexico (GOM), a new DieCAST ocean model, which uses a modified Arakawa ‘a’ grid, and the SOMS model, which uses an Arakawa ‘c’ grid, give remarkably similar results. The new model avoids ‘null space’ problems of the standard ‘a’ grid approach by first using fourth-order interpolations to a ‘c’ grid advection velocity, then applying incompressibility to the result. Accuracy is further improved by using fourth-order pressure gradient approximations at ‘a’ grid locations. Incompressibility with general geometry is satisfied efficiently by a fast-converging iteration with a regular gteometry elliptic solver. Results are compared with satellite-measured r.m.s. sea surface elevation anomaly and detailed front structures, climatological mean thermocline and empirical orthogonal functions and other observations.     

Application of cellular automata modeling to seismic elastodynamics

This article details application of a physics-based cellular automata (CA) computational approach to model seismic events in an idealized linear-elastic medium. Application of rectangular-celled CA to the seismic problem is shown to yield discrete equations equivalent to the centered-difference finite difference (FD) approach. However, it is emphasized that the discrete equations are arrived at from the ‘bottom up’ using local rules vice ‘top-down’ discretization of global partial differential equations. A further distinction between the two methods concerns the location of stresses and its impact on boundary conditions: the CA approach assigns stresses to the cell faces while the FD approach assigns stress collocated with displacement components at a single node. These differences may provide important perspective on modeling arbitrary geometry with a finite difference-like approach based on cell assembly, similar to finite element analysis. Implementation of the CA paradigm using autonomous, local cells fits naturally with object-oriented programming practices and lends itself readily to distributed computing. Results are provided for an example ground-shock simulation in which a differentiated Gaussian pulse acts on the surface of a linear-elastic half-space. The CA perspective suggests a simple treatment for the free-surface boundary condition. Comparison of the computed pressure, shear, and surface waves to those computed using a staggered-grid finite difference approach demonstrates very good agreement. In addition, the simulation results suggest that the CA approach may exhibit less ‘ringing’ as waves pass, and more symmetry in left-ward and right-ward moving waves. Future directions exploiting attractive attributes of the CA approach are suggested, to include large-scale simulation, multi-resolution analysis, and coupled-field modeling.     

Verification and validation of a winter driving simulator

A full vehicle dynamics simulator was constructed in SimCreator^® for the Cold Regions Research and Engineering Laboratory (CRREL) Instrumented Vehicle (CIV) and was used to investigate and validate the newly developed Vehicle Terrain Interaction (VTI) code. The VTI code replaces the tire component of the simulated vehicle, in the Driver and Motion Simulator (DMS), allowing it to report back realistic values while driving on various types of terrain surfaces such as mud, snow, ice, and pavement. The validation effort within this paper is focused on the winter (snow and ice) parts of the VTI code. The outputs from the Engineering Research and Development Center (ERDC) and the DMS VTI codes were validated through field experiments and against the North Atlantic Treaty Organization (NATO) Reference Mobility Model (NRMM). The DMS VTI code can be used with different vehicle models, providing the US. Army with a valuable asset that will allow simulation of existing or conceptual, manned or autonomous, ground vehicle performance for acquisition, planning, or training. This information, along with some basic terrain information, will allow troops to plan the fastest and most effective way of getting to a desired location, while minimizing the possibility of being delayed because of the terrain conditions.