A weak Galerkin generalized multiscale finite element method
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97 MATHEMATICS AND COMPUTING↗
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Results for “Finite Element Method”
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A p-version finite element method for steady incompressible fluid flow and convective heat transfer
A new p-version finite element formulation for steady, incompressible fluid flow and convective heat transfer problems is presented. The steady-state residual equations are obtained by considering a limiting case of the least-squares formulation for the transient problem. The method circumvents the Babuska-Brezzi condition, permitting the use of equal-order interpolation for velocity and pressure, without requiring the use of arbitrary parameters. Numerical results are presented to demonstrate the accuracy and generality of the method.
Winterscheidt, Daniel L.↗
Enhancing Least-Squares Finite Element Methods Through a Quantity-of-Interest
Here, we introduce an approach that augments least-squares finite element formulations with user-specified quantities-of-interest. The method incorporates the quantity-of-interest into the least-squares functional and inherits the global approximation properties of the standard formulation as well as increased resolution of the quantity-of-interest. We establish theoretical properties such as optimality and enhanced convergence under a set of general assumptions. Central to the approach is that it offers an element-level estimate of the error in the quantity-of-interest. As a result, we introduce an adaptive approach that yields efficient, adaptively refined approximations. Several numerical experiments for a range of situations are presented to support the theory and highlight the effectiveness of our methodology. Notably, the results show that the new approach is effective at improving the accuracy per total computational cost.
97 MATHEMATICS AND COMPUTING↗
A one-dimensional shock capturing finite element method and multi-dimensional generalizations
Multi-dimensional generalizations of a one-dimensional finite element shock capturing scheme are proposed. A scalar model problem is used to emphasize that 'preferred directions' are important in multi-dimensional applications. Schemes are developed for the two-dimensional Euler equations. One, based upon characteristics, employs the Mach lines and streamlines as preferred directions.
Hughes, T. J. R.↗
Development of an adaptive hp-version finite element method for computational optimal control
In this research effort, the usefulness of hp-version finite elements and adaptive solution-refinement techniques in generating numerical solutions to optimal control problems has been investigated. Under NAG-939, a general FORTRAN code was developed which approximated solutions to optimal control problems with control constraints and state constraints. Within that methodology, to get high-order accuracy in solutions, the finite element mesh would have to be refined repeatedly through bisection of the entire mesh in a given phase. In the current research effort, the order of the shape functions in each element has been made a variable, giving more flexibility in error reduction and smoothing. Similarly, individual elements can each be subdivided into many pieces, depending on the local error indicator, while other parts of the mesh remain coarsely discretized. The problem remains to reduce and smooth the error while still keeping computational effort reasonable enough to calculate time histories in a short enough time for on-board applications.
Hodges, Dewey H.↗
Robust Hybrid Finite Element Methods for Antennas and Microwave Circuits
One of the primary goals in this dissertation is concerned with the development of robust hybrid finite element-boundary integral (FE-BI) techniques for modeling and design of conformal antennas of arbitrary shape. Both the finite element and integral equation methods will be first overviewed in this chapter with an emphasis on recently developed hybrid FE-BI methodologies for antennas, microwave and millimeter wave applications. The structure of the dissertation is then outlined. We conclude the chapter with discussions of certain fundamental concepts and methods in electromagnetics, which are important to this study.
Gong, J.↗
Charged particle tracking through electrostatic wire meshes using the finite element method
Wire meshes are used across many disciplines to accelerate and focus charged particles, however, analytical solutions are non-exact and few codes exist which simulate the exact fields around a mesh with physical sizes. A tracking code based in Matlab-Simulink using field maps generated using finite element software has been developed which tracks electrons or ions through electrostatic wire meshes. The fields around such a geometry are presented as an analytical expression using several basic assumptions, however, it is apparent that computational calculations are required to obtain realistic values of electric potential and fields, particularly when multiple wire meshes are deployed. The tracking code is flexible in that any quantitatively describable particle distribution can be used for both electrons and ions as well as other benefits such as ease of export to other programs for analysis. The code is made freely available and physical examples are highlighted where this code could be beneficial for different applications.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Smoothness of Functions in Spaces of the Finite Element Method
We find necessary and sufficient conditions for the generalized smoothness of the coordinate functions obtained from the approximate relations. We show that for the coordinate functions the smoothness on their supports is equivalent to that on the boundaries of supports. We obtain conditions for the continuity of Courant type finite element approximations and conditions for the uniqueness of a linear space of such approximations.
97 MATHEMATICS AND COMPUTING↗
Hybrid finite element methods
The purpose of this paper is to show how Lagrange multipliers can be used with finite elements to achieve a number of desirable properties in the underlying approximation. For elliptic boundary value problems, variational principles can be developed in which all boundary conditions are natural. In fluid flow problems, one can endow the approximations with physically essential conservation laws.
Fix, G. M.↗
A Realistic Full-Scale 3D Modeling of Turning Using Coupled Smoothed Particle Hydrodynamics and Finite Element Method for Predicting Cutting Forces
Computational modelling is an effective technique for understanding the complex physics of machining. Large deformations, material separation, and high computational requirements are the key challenges faced while simulating machining. This work introduces a full-scale three-dimensional model of turning operations using a combined approach based on the Smoothed Particle Hydrodynamics (SPH) and Finite Element (FE) methods. By exploiting the advantages of each method, this approach leads to high-fidelity coupled SPH-FE machining models. Cutting forces and chip morphology are the primary results of interest. The machining models are validated with the results of turning experiments. Two-dimensional machining model underpredicts the cutting force and feed force by approximately 49% and 70%, respectively. Moreover, passive force cannot be predicted using the two-dimensional model. On the other hand, with the three-dimensional models developed in this manuscript, the difference between the total simulated force and experimentally measured force is ∼17%. The chip morphologies correlate with experiments in terms of the direction of the chip movement and the “long” continuous chips observed while turning Al 6061. This work expands the realm of machining simulations from two-dimensional orthogonal machining or sectional three-dimensional model to a full-scale realistic simulation. The encouraging simulation results show the potential to study more complex phenomena, such as machining stability and tool path modulation.
42 ENGINEERING↗
Finite elements and finite differences for transonic flow calculations
The paper reviews the chief finite difference and finite element techniques used for numerical solution of nonlinear mixed elliptic-hyperbolic equations governing transonic flow. The forms of the governing equations for unsteady two-dimensional transonic flow considered are the Euler equation, the full potential equation in both conservative and nonconservative form, the transonic small-disturbance equation in both conservative and nonconservative form, and the hodograph equations for the small-disturbance case and the full-potential case. Finite difference methods considered include time-dependent methods, relaxation methods, semidirect methods, and hybrid methods. Finite element methods include finite element Lax-Wendroff schemes, implicit Galerkin method, mixed variational principles, dual iterative procedures, optimal control methods and least squares.
Hafez, M. M.↗
Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations
A Petrov-Galerkin finite element formulation is presented for first-order hyperbolic systems of conservation laws with particular emphasis on the compressible Euler equations. Applications of the methodology are made to one- and two-dimensional steady and unsteady flows with shocks. Results obtained suggest the potential of the type of methods developed.
Hughes, T. J. R.↗
A Low-Rank QTT-based Finite Element Method for Elasticity Problems
We present an efficient and robust numerical algorithm for solving the linear elasticity problem that combines the Quantized Tensor Train format and a domain partitioning strategy. This approach makes it possible to solve the linear elasticity problem on a computational domain that is more general than a square. By integrating Z-ordering and subdomain concatenation, our method substantially decreases memory usage and achieves a notable reduction in rank compared to established Finite Element implementations like the FEniCS platform. This efficiency is maintained while still guaranteeing exponential convergence with respect to the number of degrees of freedom. This performance gain, however, requires a fundamental rethinking of how core finite element operations are implemented. This includes changes to mesh discretization, node and degree of freedom ordering, stiffness matrix and internal nodal force assembly, and the execution of algebraic matrix-vector operations. In this work, we discuss all these aspects in detail and assess the method’s performance in the numerical approximation of three representative test cases.
97 MATHEMATICS AND COMPUTING↗
Comparative Study of Finite Element Methods Using the Time-Accuracy-Size(TAS) Spectrum Analysis
We present a performance analysis appropriate for comparing algorithms using different numerical discretizations. By taking into account the total time-to-solution, numerical accuracy with respect to an error norm, and the computation rate, a cost-benefit analysis can be performed to determine which algorithm and discretization are particularly suited for an application. This work extends the performance spectrum model in for interpretation of hardware and algorithmic tradeoffs in numerical PDE simulation. As a proof-of-concept, popular finite element software packages are used to illustrate this analysis for Poisson’s equation.
97 MATHEMATICS AND COMPUTING↗
A crystal plasticity finite element method modeling of Zircaloy with hydride phases based on scanning electron micrographs
This work developed a microstructure-based finite element model to predict the stress state of alloys with second phase inclusions. Quantitative microstructural details extracted from scanning electron microscopy (SEM) images were used to generate heterogeneous microstructures including the morphology and spatial distribution of hydrides. Generation of digital microstructure was achieved through two steps of tessellations using software Neper and Matlab. The process is demonstrated using an example of Zircaloy material with secondary phases of hydrides dispersed within and stress-strain response of Zircaloy containing hydrides was predicted. The constitutive material model for Zircaloy in this study was based on crystal plasticity theory which considers the hexagonal close-packed (HCP) atomic structure of Zircaloy material. The hydrides were modeled as isotropic elasto-plastic material. A parametric study had been conducted to understand the effect of volume fraction, and lamellae thickness of the hydride phase on the mechanical properties of the overall material. Results can help designers to alter the manufacturing process to obtain the enhanced mechanical properties for components used in nuclear applications made by Zircaloy material.
Kulkarni, Shank S.↗
Challenges in Integrating Nondestructive Evaluation and Finite Element Methods for Realistic Structural Analysis
Capabilities and expertise related to the development of links between nondestructive evaluation (NDE) and finite element analysis (FEA) at Glenn Research Center (GRC) are demonstrated. Current tools to analyze data produced by computed tomography (CT) scans are exercised to help assess the damage state in high temperature structural composite materials. A utility translator was written to convert velocity (an image processing software) STL data file to a suitable CAD-FEA type file. Finite element analyses are carried out with MARC, a commercial nonlinear finite element code, and the analytical results are discussed. Modeling was established by building MSC/Patran (a pre and post processing finite element package) generated model and comparing it to a model generated by Velocity in conjunction with MSC/Patran Graphics. Modeling issues and results are discussed in this paper. The entire process that outlines the tie between the data extracted via NDE and the finite element modeling and analysis is fully described.
Abdul-Aziz, Ali↗
Assessing a mini-application as a performance proxy for a finite element method engineering application
The performance of a large-scale, production-quality science and engineering application (‘app’) is often dominated by a small subset of the code. Even within that subset, computational and data access patterns are often repeated, so that an even smaller portion can represent the performance-impacting features. If application developers, parallel computing experts, and computer architects can together identify this representative subset and then develop a small mini-application (‘miniapp’) that can capture these primary performance characteristics, then this miniapp can be used to both improve the performance of the app as well as provide a tool for co-design for the high-performance computing community. However, a critical question is whether a miniapp can effectively capture key performance behavior of an app. This study provides a comparison of an implicit finite element semiconductor device modeling app on unstructured meshes with an implicit finite element miniapp on unstructured meshes. The goal is to assess whether the miniapp is predictive of the performance of the app. Finally, single compute node performance will be compared, as well as scaling up to 16,000 cores. Results indicate that the miniapp can be reasonably predictive of the performance characteristics of the app for a single iteration of the solver on a single compute node.
97 MATHEMATICS AND COMPUTING↗
Process metallurgy simulation for metal drawing process optimization by using two-scale finite element method
We developed two-scale FE analysis procedure based on the crystallographic homogenization method by considering the hierarchical structure of poly-crystal aluminium alloy metal. It can be characterized as the combination of two-scale structure, such as the microscopic polycrystal structure and the macroscopic elastic plastic continuum. Micro polycrystal structure can be modeled as a three dimensional representative volume element (RVE). RVE is featured as by 3×3×3 eight-nodes solid finite elements, which has 216 crystal orientations. This FE analysis code can predict the deformation, strain and stress evolutions in the wire drawing processes in the macro- scales, and further the crystal texture and hardening evolutions in the micro-scale. In this study, we analyzed the texture evolution in the wire drawing processes by our two-scale FE analysis code under conditions of various drawing angles of dice. We evaluates the texture evolution in the surface and center regions of the wire cross section, and to clarify the effects of processing conditions on the texture evolution.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗