Publications

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A general-purpose algorithm for mesh optimization via node-movement, known as the Target-Matrix Paradigm, is introduced. The algorithm is general purpose in that it can be applied to a wide variety of mesh and element types, and to various commonly recurring mesh optimization problems such as shape improvement, and to more unusual problems like boundary-layer preservation with sliver removal, high-order mesh improvement, and edge-length equalization. The algorithm can be considered to be a direct optimization method in which weights are automatically constructed to enable definitions of application- specific mesh quality. The high-level concepts of the paradigm have been implemented in the Mesquite mesh improvement library, along with a number of concrete algorithms that address mesh quality issues such as those shown in the examples of the present paper. © Springer-Verlag (outside the USA) 2011.

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There is currently sparse literature on how to implement systematic and comprehensive processes for modern V&V/UQ (VU) within large computational simulation projects. Important design requirements have been identified in order to construct a viable 'system' of processes. Significant processes that are needed include discovery, accumulation, and assessment. A preliminary design is presented for a VU Discovery process that accounts for an important subset of the requirements. The design uses a hierarchical approach to set context and a series of place-holders that identify the evidence and artifacts that need to be created in order to tell the VU story and to perform assessments. The hierarchy incorporates VU elements from a Predictive Capability Maturity Model and uses questionnaires to define critical issues in VU. The place-holders organize VU data within a central repository that serves as the official VU record of the project. A review process ensures that those who will contribute to the record have agreed to provide the evidence identified by the Discovery process. VU expertise is an essential part of this process and ensures that the roadmap provided by the Discovery process is adequate. Both the requirements and the design were developed to support the Nuclear Energy Advanced Modeling and Simulation Waste project, which is developing a set of advanced codes for simulating the performance of nuclear waste storage sites. The Waste project served as an example to keep the design of the VU Discovery process grounded in practicalities. However, the system is represented abstractly so that it can be applied to other M&S projects.

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The objective of the U.S. Department of Energy Office of Nuclear Energy Advanced Modeling and Simulation Waste Integrated Performance and Safety Codes (NEAMS Waste IPSC) is to provide an integrated suite of computational modeling and simulation (M&S) capabilities to quantitatively assess the long-term performance of waste forms in the engineered and geologic environments of a radioactive-waste storage facility or disposal repository. To meet this objective, NEAMS Waste IPSC M&S capabilities will be applied to challenging spatial domains, temporal domains, multiphysics couplings, and multiscale couplings. A strategic verification and validation (V&V) goal is to establish evidence-based metrics for the level of confidence in M&S codes and capabilities. Because it is economically impractical to apply the maximum V&V rigor to each and every M&S capability, M&S capabilities will be ranked for their impact on the performance assessments of various components of the repository systems. Those M&S capabilities with greater impact will require a greater level of confidence and a correspondingly greater investment in V&V. This report includes five major components: (1) a background summary of the NEAMS Waste IPSC to emphasize M&S challenges; (2) the conceptual foundation for verification, validation, and confidence assessment of NEAMS Waste IPSC M&S capabilities; (3) specifications for the planned verification, validation, and confidence-assessment practices; (4) specifications for the planned evidence information management system; and (5) a path forward for the incremental implementation of this V&V plan.

Predictive Capability Maturity Model (PCMM) is a communication tool that must include a dicussion of the supporting evidence. PCMM is a tool for managing risk in the use of modeling and simulation. PCMM is in the service of organizing evidence to help tell the modeling and simulation (M&S) story. PCMM table describes what activities within each element are undertaken at each of the levels of maturity. Target levels of maturity can be established based on the intended application. The assessment is to inform what level has been achieved compared to the desired level, to help prioritize the VU activities & to allocate resources.

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Various aspects of mesh quality are surveyed to clarify the disconnect between the traditional uses of mesh quality metrics within industry and the fact that quality ultimately depends on the solution to the physical problem. Truncation error analysis for ffnite difference methods reveals no clear connection to most traditional mesh quality metrics. Finite element bounds to the interpolation error can be shown, in some cases, to be related to known quality metrics such as the condition number. On the other hand, the use of quality metrics that do not take solution characteristics into account can be valid in certain circumstances, primarily as a means of automatically detecting defective meshes. The use of such metrics when applied to simulations for which quality is highly-dependent on the physical solution is clearly inappropriate. Various ffaws and problems with existing quality metrics are mentioned, along with a discussion on the use of threshold values. In closing, the author advocates the investigation of explicitly-referenced quality metrics as a potential means of bridging the gap between a priori quality metrics and solution-dependent metrics.

Verdict is a collection of subroutines for evaluating the geometric qualities of triangles, quadrilaterals, tetrahedra, and hexahedra using a variety of functions. A quality is a real number assigned to one of these shapes depending on its particular vertex coordinates. These functions are used to evaluate the input to finite element, finite volume, boundary element, and other types of solvers that approximate the solution to partial differential equations defined over regions of space. This article describes the most recent version of Verdict and provides a summary of the main properties of the quality functions offered by the library. It finally demonstrates the versatility and applicability of Verdict by illustrating its use in several scientific applications that pertain to pre, post, and end-to-end processing.

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We present a mesh optimization algorithm for adaptively improving the finite element interpolation of a function of interest. The algorithm minimizes an objective function by swapping edges and moving nodes. Numerical experiments are performed on model problems. The results illustrate that the mesh optimization algorithm can reduce the W1,∞ semi-norm of the interpolation error. For these examples, the L2, L∞, and H1 norms decreased also.

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Verdict is a collection of subroutines for evaluating the geometric qualities of triangles, quadrilaterals, tetrahedra, and hexahedra using a variety of metrics. A metric is a real number assigned to one of these shapes depending on its particular vertex coordinates. These metrics are used to evaluate the input to finite element, finite volume, boundary element, and other types of solvers that approximate the solution to partial differential equations defined over regions of space. The geometric qualities of these regions is usually strongly tied to the accuracy these solvers are able to obtain in their approximations. The subroutines are written in C++ and have a simple C interface. Each metric may be evaluated individually or in combination. When multiple metrics are evaluated at once, they share common calculations to lower the cost of the evaluation.

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We compare inexact Newton and coordinate descent optimization methods for improving the quality of a mesh by repositioning the vertices, where the overall quality is measured by the harmonic mean of the mean-ratio metric. The effects of problem size, element size heterogeneity, and various vertex displacement schemes on the performance of these algorithms are assessed for a series of tetrahedral meshes.

Order-of-accuracy verification is necessary to ensure that software correctly solves a given set of equations. One method to verify the order of accuracy of a code is the method of manufactured solutions. In this study, a manufactured solution has been derived and implemented that allows verification of not only the Euler, Navier-Stokes, and Reynolds-Averaged Navier-Stokes (RANS) equation sets, but also some of their associated boundary conditions (BC's): slip, no-slip (adiabatic and isothermal), and outflow (subsonic, supersonic, and mixed). Order-of-accuracy verification has been performed for the Euler and Navier-Stokes equations and these BC's in a compressible computational fluid dynamics code. All of the results shown are on skewed, non-uniform meshes. RANS results will be presented in a future paper. The observed order of accuracy was lower than the expected order of accuracy in two cases. One of these cases resulted in the identification and correction of a coding mistake in the CHAD gradient correction that was reducing the observed order of accuracy. This mistake would have been undetectable on a Cartesian mesh. During the search for the CHAD gradient correction problem, an unrelated coding mistake was found and corrected. The other case in which the observed order of accuracy was less than expected was a test of the slip BC; although no specific coding or formulation mistakes have yet been identified. After the correction of the identified coding mistakes, all of the aforementioned equation sets and BC's demonstrated the expected (or at least acceptable) order of accuracy except the slip condition.

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We compare inexact Newton and coordinate descent methods for optimizing the quality of a mesh by repositioning the vertices, where quality is measured by the harmonic mean of the mean-ratio metric. The effects of problem size, element size heterogeneity, and various vertex displacement schemes on the performance of these algorithms are assessed for a series of tetrahedral meshes.

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We present a new shape measure for tetrahedral elements that is optimal in that it gives the distance of a tetrahedron from the set of inverted elements. This measure is constructed from the condition number of the linear transformation between a unit equilateral tetrahedron and any tetrahedron with positive volume. Using this shape measure, we formulate two optimization objective functions that are differentiated by their goal: the first seeks to improve the average quality of the tetrahedral mesh; the second aims to improve the worst-quality element in the mesh. We review the optimization techniques used with each objective function and present experimental results that demonstrate the effectiveness of the mesh improvement methods. We show that a combined optimization approach that uses both objective functions obtains the best-quality meshes for several complex geometries. Copyright © 2001 John Wiley and Sons, Ltd.

We investigate a well-motivated mesh untangling objective function whose optimization automatically produces non-inverted elements when possible. Examples show the procedure is highly effective on simplicial meshes and on non-simplicial (e.g., hexahedral) meshes constructed via mapping or sweeping algorithms. The current whisker-weaving (WW) algorithm in CUBIT usually produces hexahedral meshes that are unsuitable for analyses due to inverted elements. The majority of these meshes cannot be untangled using the new objective function. The most likely source of the difficulty is poor mesh topology.

Sweeping has become the workhorse algorithm for creating conforming hexahedral meshes of complex models. This paper describes progress on the automatic, robust generation of MultiSwept meshes in CUBIT. MultiSweeping extends the class of volumes that may be swept to include those with multiple source and multiple target surfaces. While not yet perfect, CUBIT's MultiSweeping has recently become more reliable, and been extended to assemblies of volumes. Sweep Forging automates the process of making a volume (multi) sweepable: Sweep Verification takes the given source and target surfaces, and automatically classifies curve and vertex types so that sweep layers are well formed and progress from sources to targets.

A procedure for code Verification by the Method of Manufactured Solutions (MMS) is presented. Although the procedure requires a certain amount of creativity and skill, we show that MMS can be applied to a variety of engineering codes which numerically solve partial differential equations. This is illustrated by detailed examples from computational fluid dynamics. The strength of the MMS procedure is that it can identify any coding mistake that affects the order-of-accuracy of the numerical method. A set of examples which use a blind-test protocol demonstrates the kinds of coding mistakes that can (and cannot) be exposed via the MMS code Verification procedure. The principle advantage of the MMS procedure over traditional methods of code Verification is that code capabilities are tested in full generality. The procedure thus results in a high degree of confidence that all coding mistakes which prevent the equations from being solved correctly have been identified.

Quality metrics for structured and unstructured mesh generation are placed within an algebraic framework to form a mathematical theory of mesh quality metrics. The theory, based on the Jacobian and related matrices, provides a means of constructing, classifying, and evaluating mesh quality metrics. The Jacobian matrix is factored into geometrically meaningful parts. A nodally-invariant Jacobian matrix can be defined for simplicial elements using a weight matrix derived from the Jacobian matrix of an ideal reference element. Scale and orientation-invariant algebraic mesh quality metrics are defined. the singular value decomposition is used to study relationships between metrics. Equivalence of the element condition number and mean ratio metrics is proved. Condition number is shown to measure the distance of an element to the set of degenerate elements. Algebraic measures for skew, length ratio, shape, volume, and orientation are defined abstractly, with specific examples given. Combined metrics for shape and volume, shape-volume-orientation are algebraically defined and examples of such metrics are given. Algebraic mesh quality metrics are extended to non-simplical elements. A series of numerical tests verify the theoretical properties of the metrics defined.