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NASA NTRS2003

A Computationally Efficient Meshless Local Petrov-Galerkin Method for Axisymmetric Problems

The Meshless Local Petrov-Galerkin (MLPG) method is one of the recently developed element-free methods. The method is convenient and can produce accurate results with continuous secondary variables, but is more computationally expensive than the finite element method. To overcome this disadvantage, a simple Heaviside test function is chosen. The computational effort is significantly reduced by eliminating the domain integral for the axisymmetric potential problems and by simplifying the domain integral for the axisymmetric elasticity problems. The method is evaluated through several patch tests for axisymmetric problems and example problems for which the exact solutions are available. The present method yielded very accurate solutions. The sensitivity of several parameters of the method is also studied.

Raju, I. S.↗

DOE OSTI2015

Laser shock peening on Zr-based bulk metallic glass and its effect on plasticity: Experiment and modeling

The Zr-based bulk metallic glasses (BMGs) are a new family of attractive materials with good glass-forming ability and excellent mechanical properties, such as high strength and excellent wear resistance, which make them candidates for structural and biomedical materials. Although the mechanical behavior of BMGs has been widely investigated, their deformation mechanisms are still poorly understood. In particular, their poor ductility significantly impedes their industrial application. In the present work, we show that the ductility of Zr-based BMGs with nearly zero plasticity is improved by a laser shock peening technique. Moreover, we map the distribution of laser-induced residual stresses via the micro-slot cutting method, and then predict them using a three dimensional finite-element method coupled with a confined plasma model. Reasonable agreement is achieved between the experimental and modeling results. The analysis of serrated flow reveals plentiful and useful information of the underlying deformation process. As a result, our work provides an easy and effective way to extend the ductility of intrinsically-brittle BMGs, opening up wider applications of these materials.

36 MATERIALS SCIENCE↗

DOE OSTI2014

Microwave gas breakdown in elliptical waveguides

This paper analyzes the microwave gas discharge within elliptical waveguides excited by the fundamental mode. The Rayleigh-Ritz method has been applied to solve the continuity equation. The eigenvalue problem defined by the breakdown condition has been solved and the effective diffusion length of the elliptical waveguide has been calculated, what is used to find the corona threshold. This paper extends the microwave breakdown model developed for circular waveguides and shows the better corona withstanding capabilities of elliptical waveguides. The corona breakdown electric field threshold obtained with the variational method has been compared with the one calculated with the Finite Elements Method, showing excellent agreement.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2016

Multiscale analysis and computation for flows in heterogeneous media

Our work in this project is aimed at making fundamental advances in multiscale methods for flow and transport in highly heterogeneous porous media. The main thrust of this research is to develop a systematic multiscale analysis and efficient coarse-scale models that can capture global effects and extend existing multiscale approaches to problems with additional physics and uncertainties. A key emphasis is on problems without an apparent scale separation. Multiscale solution methods are currently under active investigation for the simulation of subsurface flow in heterogeneous formations. These procedures capture the effects of fine-scale permeability variations through the calculation of specialized coarse-scale basis functions. Most of the multiscale techniques presented to date employ localization approximations in the calculation of these basis functions. For some highly correlated (e.g., channelized) formations, however, global effects are important and these may need to be incorporated into the multiscale basis functions. Other challenging issues facing multiscale simulations are the extension of existing multiscale techniques to problems with additional physics, such as compressibility, capillary effects, etc. In our project, we explore the improvement of multiscale methods through the incorporation of additional (single-phase flow) information and the development of a general multiscale framework for flows in the presence of uncertainties, compressible flow and heterogeneous transport, and geomechanics. We have considered (1) adaptive local-global multiscale methods, (2) multiscale methods for the transport equation, (3) operator-based multiscale methods and solvers, (4) multiscale methods in the presence of uncertainties and applications, (5) multiscale finite element methods for high contrast porous media and their generalizations, and (6) multiscale methods for geomechanics. Below, we present a brief overview of each of these contributions.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2016

Adaptive Mixed GMsFEM for Flows in Heterogeneous Media

Abstract In this paper, we present two adaptive methods for the basis enrichment of the mixed Generalized Multiscale Finite Element Method (GMsFEM) for solving the flow problem in heterogeneous media. We develop an a-posteriori error indicator which depends on the norm of a local residual operator. Based on this indicator, we construct an offline adaptive method to increase the number of basis functions locally in coarse regions with large local residuals. We also develop an online adaptive method which iteratively enriches the function space by adding new functions computed based on the residual of the previous solution and special minimum energy snapshots. We show theoretically and numerically the convergence of the two methods. The online method is, in general, better than the offline method as the online method is able to capture distant effects (at a cost of online computations), and both methods have faster convergence than a uniform enrichment. Analysis shows that the online method should start with a certain number of initial basis functions in order to have the best performance. The numerical results confirm this and show further that with correct selection of initial basis functions, the convergence of the online method can be independent of the contrast of the medium. We consider cases with both very high and very low conducting inclusions and channels in our numerical experiments.

Mathematics↗

NASA NTRS1975

Finite element computation of a viscous compressible free shear flow governed by the time dependent Navier-Stokes equations

A finite element algorithm for solution of fluid flow problems characterized by the two-dimensional compressible Navier-Stokes equations was developed. The program is intended for viscous compressible high speed flow; hence, primitive variables are utilized. The physical solution was approximated by trial functions which at a fixed time are piecewise cubic on triangular elements. The Galerkin technique was employed to determine the finite-element model equations. A leapfrog time integration is used for marching asymptotically from initial to steady state, with iterated integrals evaluated by numerical quadratures. The nonsymmetric linear systems of equations governing time transition from step-to-step are solved using a rather economical block iterative triangular decomposition scheme. The concept was applied to the numerical computation of a free shear flow. Numerical results of the finite-element method are in excellent agreement with those obtained from a finite difference solution of the same problem.

Cooke, C. H.↗

NASA NTRS1982

Development of a reduced basis technique for transient thermal analysis

A technique to reduce the degrees of freedom in static and dynamic problems, the reduced basis method, is described. The method combines the classical Rayleigh-Ritz approximation with contemporary finite element methods to retain modeling versatility as the degrees of freedom are reduced. Applications to a nonlinear dynamic response problem are discussed efforts to apply the method to nonlinear transient thermal response problems are summarized. The selection of basis vectors for reducing the system of equations is addressed.

Shore, C. P.↗

NASA NTRS1987

A general method for dynamic analysis of structures overview

The presented research deals with the development of a dynamic analysis method for structural systems. The modeling approach is essentially a finite element method in the sense that the structure is divided into n elements. An element is defined as any structural unit whose degree of freedom (dofs) can be categorized as either interface or non-interface dofs. An element could be a fundamental unit such as a rod, a beam, a plate etc., or it could be an entire structural component. Furthermore, the parameters for the element could be distributed or lumped. The choice of elements is totally arbitrary and is a matter of user convenience. In particular, issues of accuracy and convergence do not enter on the level of example that bookkeeping is reduced to a minimum. Each element is modeled using a set of interface constraint modes (ICM) combined with a set of interface restrained normal models (IRNM). The next step is the solution of the system eigenvalue problem. The procedure calls for the sequential solution of a number of small eigenvalue problems based on a truncation principle for IRNM. In addition, the form of these eigenvalue problems is very simple such that an escalator type of eigenvalue problem solver can be used which is extremely cost-effective and fast.

Engels, Remi C.↗

NASA NTRS2003

Three-Dimensional Magnetic Analysis Technique Developed for Evaluating Stirling Convertor Linear Alternators

The Department of Energy, the Stirling Technology Company (STC), and the NASA Glenn Research Center are developing Stirling convertors for Stirling radioisotope generators to provide electrical power for future NASA deep space missions. STC is developing the 55-We technology demonstration convertor (TDC) under contract to the Department of Energy. The Department of Energy recently named Lockheed Martin as the system integration contractor for the Stirling radioisotope generator development project. Lockheed Martin will develop the Stirling radioisotope generator engineering unit and has contract options to develop the qualification unit and the first flight unit. Glenn s role includes an in-house project to provide convertor, component, and materials testing and evaluation in support of the overall power system development. As a part of this work, Glenn has established an in-house Stirling research laboratory for testing, analyzing, and evaluating Stirling machines. STC has built four 55-We convertors for NASA, and these are being tested at Glenn. A cross-sectional view of the 55-We TDC is shown in the figure. Of critical importance to the successful development of the Stirling convertor for space power applications is the development of a lightweight and highly efficient linear alternator. In support, Glenn has been developing finite element analysis and finite element method tools for performing various linear alternator thermal and electromagnetic analyses and evaluating design configurations. A three-dimensional magnetostatic finite element model of STC's 55-We TDC linear alternator was developed to evaluate the demagnetization fields affecting the alternator magnets. Since the actual linear alternator hardware is symmetric to the quarter section about the axis of motion, only a quarter section of the alternator was modeled. The components modeled included the mover laminations, the neodymium-iron-boron magnets, the stator laminations, and the copper coils. The three-dimensional magnetostatic model was then coupled with a circuit simulator model of the alternator load and convertor controller. The coupled model was then used to generate alternator terminal voltage and current predictions. The predicted voltage and current waveforms agreed well with the experimental data, which tended to validate the accuracy of the coupled model. The model was then used to generate predictions of the demagnetization fields acting on the alternator magnets for the alternator under load. The preliminary model predictions indicate that the highest potential for demagnetization is along the inside surface of the uncovered magnets. The demagnetization field for the uncovered magnets when the mover is positioned at the end of a stroke is higher than it is when the mover is at the position of maximum induced voltage or maximum alternator current. Assuming normal load conditions, the model predicted that the onset of demagnetization is most likely to occur for magnet temperatures above 101 C.

Geng, Steven M.↗

DOE OSTI2014

Edge Equilibrium Code (EEC) For Tokamaks

The edge equilibrium code (EEC) described in this paper is developed for simulations of the near edge plasma using the finite element method. It solves the Grad-Shafranov equation in toroidal coordinate and uses adaptive grids aligned with magnetic field lines. Hermite finite elements are chosen for the numerical scheme. A fast Newton scheme which is the same as implemented in the equilibrium and stability code (ESC) is applied here to adjust the grids.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS2000

Determination of the Fracture Parameters in a Stiffened Composite Panel

A modified J-integral, namely the equivalent domain integral, is derived for a three-dimensional anisotropic cracked solid to evaluate the stress intensity factor along the crack front using the finite element method. Based on the equivalent domain integral method with auxiliary fields, an interaction integral is also derived to extract the second fracture parameter, the T-stress, from the finite element results. The auxiliary fields are the two-dimensional plane strain solutions of monoclinic materials with the plane of symmetry at x(sub 3) = 0 under point loads applied at the crack tip. These solutions are expressed in a compact form based on the Stroh formalism. Both integrals can be implemented into a single numerical procedure to determine the distributions of stress intensity factor and T-stress components, T11, T13, and thus T33, along a three-dimensional crack front. The effects of plate thickness and crack length on the variation of the stress intensity factor and T-stresses through the thickness are investigated in detail for through-thickness center-cracked plates (isotropic and orthotropic) and orthotropic stiffened panels under pure mode-I loading conditions. For all the cases studied, T11 remains negative. For plates with the same dimensions, a larger size of crack yields larger magnitude of the normalized stress intensity factor and normalized T-stresses. The results in orthotropic stiffened panels exhibit an opposite trend in general. As expected, for the thicker panels, the fracture parameters evaluated through the thickness, except the region near the free surfaces, approach two-dimensional plane strain solutions. In summary, the numerical methods presented in this research demonstrate their high computational effectiveness and good numerical accuracy in extracting these fracture parameters from the finite element results in three-dimensional cracked solids.

Lin, Chung-Yi↗

DOE OSTI2015

Liquid crystal free energy relaxation by a theoretically informed Monte Carlo method using a finite element quadrature approach

A theoretically informed Monte Carlo method is proposed for Monte Carlo simulation of liquid crystals on the basis of theoretical representations in terms of coarse-grained free energy functionals. The free energy functional is described in the framework of the Landau-de Gennes formalism. A piecewise finite element discretization is used to approximate the alignment field, thereby providing an excellent geometrical representation of curved interfaces and accurate integration of the free energy. The method is suitable for situations where the free energy functional includes highly non-linear terms, including chirality or high-order deformation modes. Finally, the validity of the method is established by comparing the results of Monte Carlo simulations to traditional Ginzburg-Landau minimizations of the free energy using a finite difference scheme, and its usefulness is demonstrated in the context of simulations of chiral liquid crystal droplets with and without nanoparticle inclusions.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS1986

Least squares finite element simulation of transonic flows

Finite difference approximation of transonic flow problems is a well-developed and largely successful approach. Nevertheless, there is still a real need to develop finite element methods for applications arising from fluid-structure interactions and problems with complicated boundaries. In this paper a least squares based finite element scheme is introduced. It is shown that, if suitably formulated, such an approach can lead to physically meaningful results. Bottlenecks that arise from such schemes are also discussed.

Chen, T. F.↗

DOE OSTI2020

Efficient Low-Order Refined Preconditioners for High-Order Matrix-Free Continuous and Discontinuous Galerkin Methods

In this paper, we design preconditioners for the matrix-free solution of high-order continuous and discontinuous Galerkin discretizations of elliptic problems based on finite element method--spectral element method (FEM-SEM) equivalence and additive Schwarz methods. The high-order operators are applied without forming the system matrix, making use of sum factorization for efficient evaluation. The system is preconditioned using a spectrally equivalent low-order (p = 1) finite element operator discretization on a refined mesh. The low-order refined mesh is anisotropic and not shape regular in the polynomial degree of the high-order operator, requiring specialized solvers to treat the anisotropy. We make use of an element-structured, geometric multigrid V-cycle with ordered ILU(0) smoothing. The preconditioner is parallelized through an overlapping additive Schwarz method that is robust in h and p. The method is extended to interior penalty and Bassi and Rebay (BR2) discontinuous Galerkin discretizations, for which it is also robust in the size of the penalty parameter. Finally, numerical results are presented on a variety of examples, verifying the uniformity of the preconditioner.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2015

Three-dimensional local ALE-FEM method for fluid flow in domains containing moving boundaries/objects interfaces

A three-dimensional finite element method for the numerical simulations of fluid flow in domains containing moving rigid objects or boundaries is developed. The method falls into the general category of Arbitrary Lagrangian Eulerian methods; it is based on a fixed mesh that is locally adapted in the immediate vicinity of the moving interfaces and reverts to its original shape once the moving interfaces go past the elements. The moving interfaces are defined by separate sets of marker points so that the global mesh is independent of interface movement and the possibility of mesh entanglement is eliminated. The results is a fully robust formulation capable of calculating on domains of complex geometry with moving boundaries or devises that can also have a complex geometry without danger of the mesh becoming unsuitable due to its continuous deformation thus eliminating the need for repeated re-meshing and interpolation. Moreover, the boundary conditions on the interfaces are imposed exactly. This work is intended to support the internal combustion engines simulator KIVA developed at Los Alamos National Laboratories. The model's capabilities are illustrated through application to incompressible flows in different geometrical settings that show the robustness and flexibility of the technique to perform simulations involving moving boundaries in a three-dimensional domain.

16 TIDAL AND WAVE POWER↗

NASA NTRSDate not supplied

A Moving Discontinuous Galerkin Method with Interface Condition Enforcement for Reacting Hypersonic Flows

The necessity of enforcing conservation in computational elements or cells (element conservation) for discontinuous solutions is well understood and respected for solving conservation laws in computational fluid dynamics (CFD). In contrast, interface conservation, where the conservation across cell interfaces is enforced, is long ignored, and yet is also ruled and required by the underlying physics just like element conservation. Violation of the interface conservation across discontinuities is the root cause why an exact discontinuous solution can never be achieved in shock capturing methods. The interface conservation is examined and explored in this talk. Moving discontinuous Galerkin (MDG) finite element method with interface conservation enforcement (MDG-ICE) [1],[2] are then presented for solving compressible flow problems with discontinuities based on the observation that the interface conservation can only be satisfied, only when mesh interfaces are aligned with discontinuities. In the MDG-ICE formulation, both conservative quantities and grid geometry are considered as independent variables. A space-time DG formulation is used to solve the multi-material compressible Euler equations in the standard discontinuous solution space and the discrete grid geometry is solved using a variational formulation in a continuous space. A self-adaptive Levenberg-Marquardt method is utilized to solve the resulting over-determined system of nonlinear equations arising from the MDG-ICE formulation. A number of numerical experiments for a variety of flow problems are conducted to assess the accuracy and performance of the MDG-ICE method. Numerical results obtained indicate that the MDG-ICE method is able to deliver the designed order of both h- and p-convergence even for discontinuous solutions, and detect all types of interfaces, via interface condition enforcement and satisfy, via grid movement, the compressible Euler equations and the associated interface condition.

Discontinuous Galekin↗

NASA NTRS1981

Optimum structural design with plate bending elements - A survey

A survey is presented of recently published papers in the field of optimum structural design of plates, largely with respect to the minimum-weight design of plates subject to such constraints as fundamental frequency maximization. It is shown that, due to the availability of powerful computers, the trend in optimum plate design is away from methods tailored to specific geometry and loads and toward methods that can be easily programmed for any kind of plate, such as finite element methods. A corresponding shift is seen in optimization from variational techniques to numerical optimization algorithms. Among the topics covered are fully stressed design and optimality criteria, mathematical programming, smooth and ribbed designs, design against plastic collapse, buckling constraints, and vibration constraints.

Haftka, R. T.↗

DOE OSTI2025

Gradient flow based phase-field modeling using separable neural networks

Allen–Cahn equation is a reaction–diffusion equation and is widely used for modeling phase separation. Machine learning methods for solving the Allen–Cahn equation in its strong form suffer from inaccuracies in collocation techniques, errors in computing higher-order spatial derivatives, and the large system size required by the space–time approach. To overcome these challenges, we propose solving the gradient flow of the Ginzburg–Landau free energy functional, which is equivalent to the Allen–Cahn equation, thereby avoiding the second-order spatial derivatives associated with the Allen–Cahn equation. A minimizing movement scheme is employed to solve the gradient flow problem, eliminating the complexities of a space–time approach. We utilize a separable neural network that efficiently represents the phase field through low-rank tensor decomposition. As we use the minimizing movement scheme to numerically solve the gradient flow problem, we thus, refer to the proposed method as the Separable Deep Minimizing Movement (SDMM) method. The evaluation of the functional in the minimizing movement scheme using the Gauss quadrature technique bypasses the inaccuracies associated with collocation techniques traditionally used to solve partial differential equations. A hyperbolic tangent transformation is introduced on the phase field prior to the evaluation of the functional to ensure that it remains strictly bounded within the values of the two phases. For this transformation, theoretical guarantee for energy stability of the minimizing movement scheme is established. Our results suggest that this transformation helps to improve the accuracy and efficiency significantly. The proposed method resolves the challenges faced by state-of-the-art machine learning techniques, outperforming them in both accuracy and efficiency. It is also the first machine learning method to achieve an order of magnitude speed improvement over the finite element method. In addition to its formulation and computational implementation, several case studies illustrate the applicability of the proposed method.

42 ENGINEERING↗