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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

The eXtended virtual element method for elliptic problems with weakly singular solutions

This paper introduces a novel eXtended virtual element method, an extension of the conforming virtual element method. The X-VEM is formulated by incorporating appropriate enrichment functions in the local spaces. The method is designed to handle highly generic enrichment functions, including singularities arising from fractured domains. By achieving consistency on the enrichment space, the method is proven to achieve arbitrary approximation orders even in the presence of singular solutions. The paper includes a complete convergence analysis under general assumptions on mesh regularity, and numerical experiments validating the method’s accuracy on various mesh families, demonstrating optimal convergence rates in the L 2 - and H 1 - norms on fractured or L-shaped domains.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Numerical integration in the virtual element method with the scaled boundary cubature scheme

Abstract The virtual element method (VEM) is a stabilized Galerkin method on meshes that consist of arbitrary (convex and nonconvex) polygonal and polyhedral elements. A crucial ingredient in the implementation of low‐ and high‐order VEM is the numerical integration of monomials and nonpolynomial functions over such elements. In this article, we apply the recently proposed scaled boundary cubature (SBC) scheme to compute the weak form integrals in various virtual element formulations over polygonal and polyhedral meshes. In doing so, we demonstrate the flexibility of the approach and the accuracy that it delivers on a broad suite of boundary‐value problems in 2D and 3D over polytopes with affine faces as well as on elements with curved boundaries. In addition, the use of the SBC scheme is exemplified in an enriched Poisson formulation of the VEM in which weakly singular functions are required to be integrated. This study establishes the SBC method as a simple, accurate and efficient integration scheme for use in the VEM.

Chin, Eric B.↗
DOE OSTI2024

Stress-hybrid virtual element method on six-noded triangular meshes for compressible and nearly-incompressible linear elasticity

In this paper, we present a first-order Stress-Hybrid Virtual Element Method (SH-VEM) on six-noded triangular meshes for linear plane elasticity. Here, we adopt the Hellinger–Reissner variational principle to construct a weak equilibrium condition and a stress based projection operator. In each element, the stress projection operator is expressed in terms of the nodal displacements, which leads to a displacement based formulation. This stress-hybrid approach assumes a globally continuous displacement field while the stress field is discontinuous across each element. The stress field is initially represented by divergence-free tensor polynomials based on Airy stress functions, but we also present a formulation that uses a penalty term to enforce the element equilibrium conditions, referred to as the Penalty Stress-Hybrid Virtual Element Method (PSH-VEM). Numerical results are presented for PSH-VEM and SH-VEM, and we compare their convergence to the composite triangle FEM and B-bar VEM on benchmark problems in linear elasticity. The SH-VEM converges optimally in the L 2 norm of the displacement, energy seminorm, and the L 2 norm of hydrostatic stress. Furthermore, the results reveal that PSH-VEM converges in most cases at a faster rate than the expected optimal rate, but it requires the selection of a suitably chosen penalty parameter.

42 ENGINEERING↗
DOE OSTI2024

A $C^1$-Conforming Arbitrary-Order Two-Dimensional Virtual Element Method for the Fourth-Order Phase-Field Equation

We present a two-dimensional conforming virtual element method for the fourth-order phase-field equation. Our proposed numerical approach to the solution of this high-order phase-field (HOPF) equation relies on the design of an arbitrary-order accurate, virtual element space with $C^1$ global regularity. Such regularity is guaranteed by taking the values of the virtual element functions and their full gradient at the mesh vertices as degrees of freedom. Attaining high-order accuracy requires also edge polynomial moments of the trace of the virtual element functions and their normal derivatives. In this work, we detail the scheme construction, and prove its convergence by deriving error estimates in different norms. A set of representative test cases allows us to assess the behavior of the method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

A guide to the design of the virtual element methods for second- and fourth-order partial differential equations

Here we discuss the design and implementation details of two conforming virtual element methods for the numerical approximation of two partial differential equations that emerge in phase-field modeling of fracture propagation in elastic material. The two partial differential equations are: (i) a linear hyperbolic equation describing the momentum balance and (ii) a fourth-order elliptic equation modeling the damage of the material. Inspired by, we develop a new conforming VEM for the discretization of the two equations, which is implementation-friendly, i.e., different terms can be implemented by exploiting a single projection operator. We use C 0 and C 1 virtual elements for the second-and fourth-order partial differential equation, respectively. For both equations, we review the formulation of the virtual element approximation and discuss the details pertaining the implementation.

42 ENGINEERING↗
DOE OSTI2024

Arbitrary Order Virtual Element Methods for High‐Order Phase‐Field Modeling of Dynamic Fracture

ABSTRACT Accurate modeling of fracture nucleation and propagation in brittle and ductile materials subjected to dynamic loading is important in predicting material damage and failure under extreme conditions. Phase‐field fracture models have garnered a lot of attention in recent years due to their success in representing damage and fracture processes in a wide class of materials and under a variety of loading conditions. Second‐order phase‐field fracture models are by far the most popular among researchers (and increasingly, among practitioners), but fourth‐order models have started to gain broader acceptance since their more recent introduction. The exact solution corresponding to these high‐order phase‐field fracture models has higher regularity. Thus, numerical solutions of the model equations can achieve improved accuracy and higher spatial convergence rates. In this work, we develop a virtual element framework for the high‐order phase‐field model of dynamic fracture. The virtual element method (VEM) can be regarded as a generalization of the classical finite element method. In addition to many other desirable characteristics, the VEM allows computing on polytopal meshes. Here, we use ‐conforming virtual elements and the generalized‐ time integration method for the momentum balance equation, and adopt ‐conforming virtual elements for the high‐order phase‐field equation. We verify our virtual element framework using classical quasi‐static benchmark problems and demonstrate its capabilities with the aid of numerical simulations of dynamic fracture in brittle materials.

42 ENGINEERING↗
DOE OSTI2025

Virtual element approximations of the time-fractional nonlinear convection-diffusion equation on polygonal meshes

We extend the Virtual Element Method to a two-dimensional unsteady nonlinear convection-diffusion equation characterized by a fractional-order derivative with respect to the time variable. Our methodology is based on three fundamental technical components: a fractional version of the Grunwald-Letnikov approximation, discrete maximal regularity, and the regularity theory associated with non-linearity. We prove the method's well-posedness, i.e., the approximate solution's existence and uniqueness to the time-fractional convection-diffusion equation with a Lipschitz nonlinear source term. The fully discrete scheme inherently maintains stability and consistency by leveraging the discrete maximal regularity and the energy projection operator. The convergence in the L 2 -norm and H 1 -norm to various mesh configurations is validated by numerical results, underlining the practical effectiveness of the proposed method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Hourglass control in staggered-grid hydrodynamics using virtual element stabilization techniques

Numerical simulations using the staggered-grid hydrodynamics (SGH) discretization suffer from hourglass instabilities. In this work, we develop a stabilization method to suppress the hourglass instabilities using techniques from the virtual element method (VEM). The stiffness matrix of the VEM consists of two terms: the consistency matrix which is rank deficient and the stability matrix. Here, we first show that in two dimensions and on general polygons, the stiffness matrix of the SGH is identical to the consistency matrix of the linear VEM for both the diffusion equation and the linear elasticity equation. These analyses explain the origin of the hourglass instabilities of the SGH discretization method, and establish a theoretical foundation for our proposed stabilization method by augmenting the stiffness matrix of the SGH discretization using the VEM stability matrix. Then, we present numerical examples using Lagrangian SGH simulations. The numerical experiments demonstrate that the proposed VEM stabilization method is effective at eliminating hourglass modes in the SGH discretization.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2001

Finite Element Methods for real-time Haptic Feedback of Soft-Tissue Models in Virtual Reality Simulators

We have applied the linear elastic finite element method to compute haptic force feedback and domain deformations of soft tissue models for use in virtual reality simulators. Our results show that, for virtual object models of high-resolution 3D data (>10,000 nodes), haptic real time computations (>500 Hz) are not currently possible using traditional methods. Current research efforts are focused in the following areas: 1) efficient implementation of fully adaptive multi-resolution methods and 2) multi-resolution methods with specialized basis functions to capture the singularity at the haptic interface (point loading). To achieve real time computations, we propose parallel processing of a Jacobi preconditioned conjugate gradient method applied to a reduced system of equations resulting from surface domain decomposition. This can effectively be achieved using reconfigurable computing systems such as field programmable gate arrays (FPGA), thereby providing a flexible solution that allows for new FPGA implementations as improved algorithms become available. The resulting soft tissue simulation system would meet NASA Virtual Glovebox requirements and, at the same time, provide a generalized simulation engine for any immersive environment application, such as biomedical/surgical procedures or interactive scientific applications.

Frank, Andreas O.↗
NASA NTRS1973

Nonlinear transient responses of structures by the spatial finite-element method.

Based upon the Principle of Virtual Work and D'Alembert's Principle, the assumed-displacement version of the spatial finite-element method is developed to predict the large deflection transient responses of structures including elastic-plastic, strain hardening, and strain-rate material behavior. The formulations are developed in detail for curved beamlike structures undergoing planar (1) Bernouilli-Euler-type or (2) Timoshenko-type deformation behavior. The resulting equations of motion are solved timewise by a finite-difference numerical procedure. The present predictions are evaluated via several beam and ring examples for which experimental measurements and independent finite-difference predictions in both space and time are available; very good agreement is noted. The consequences of employing several types of timewise finite-difference operators are examined. Also, some comparisons between finite-element predictions and finite-difference predictions are shown to illustrate 'typical comparisons' of efficiency for a given prediction accuracy.

Wu, R. W.-H.↗
NASA NTRSDate not supplied

Hierarchical Multiscale Process Modeling of A Textile Composite Y-Joint for the Aurora D8 Aircraft

Woven polymer matrix composites (PMCs) are widely used in aerospace applications for their favorable specific properties. However, the behavior of the inter- and intra-tow matrix during manufacturing of woven PMCs is difficult to isolate experimentally due to geometric and processing complexities. Classical closed-form homogenization approaches allow for composite thermos-chemo-mechanical properties to be characterized at any fiber volume fraction. In this work, virtual curing is applied to the inter- and intra-tow matrix materials of a plain weave repeating unit cell (RUC). Thermo-chemo-mechanical property evolution within the tows is calculated using the Concentric Cylinder Model (CCM) and the Rule of Mixtures (ROM). Curing is implemented through user-written subroutines within the commercial finite element method software Abaqus. Virtual curing results are compared with those from a linear-elastic thermal cool down simulation in Abaqus. Results indicate that tow property evolution during curing has a non-negligible effect on the final stress state.

curing↗
NASA NTRS2016

Modeling Fatigue Damage Onset and Progression in Composites Using an Element-Based Virtual Crack Closure Technique Combined With the Floating Node Method

A new methodology is proposed to model the onset and propagation of matrix cracks and delaminations in carbon-epoxy composites subject to fatigue loading. An extended interface element, based on the Floating Node Method, is developed to represent delaminations and matrix cracks explicitly in a mesh independent fashion. Crack propagation is determined using an element-based Virtual Crack Closure Technique approach to determine mixed-mode energy release rates, and the Paris-Law relationship to obtain crack growth rate. Crack onset is determined using a stressbased onset criterion coupled with a stress vs. cycle curve and Palmgren-Miner rule to account for fatigue damage accumulation. The approach is implemented in Abaqus/Standard® via the user subroutine functionality. Verification exercises are performed to assess the accuracy and correct implementation of the approach. Finally, it was demonstrated that this approach captured the differences in failure morphology in fatigue for two laminates of identical stiffness, but with layups containing θdeg plies that were either stacked in a single group, or distributed through the laminate thickness.

De Carvalho, Nelson V.↗
NASA NTRS2013

Fracture Mechanics Analyses for Interface Crack Problems - A Review

Recent developments in fracture mechanics analyses of the interfacial crack problem are reviewed. The intent of the review is to renew the awareness of the oscillatory singularity at the crack tip of a bimaterial interface and the problems that occur when calculating mode mixity using numerical methods such as the finite element method in conjunction with the virtual crack closure technique. Established approaches to overcome the nonconvergence issue of the individual mode strain energy release rates are reviewed. In the recent literature many attempts to overcome the nonconvergence issue have been developed. Among the many approaches found only a few methods hold the promise of providing practical solutions. These are the resin interlayer method, the method that chooses the crack tip element size greater than the oscillation zone, the crack tip element method that is based on plate theory and the crack surface displacement extrapolation method. Each of the methods is validated on a very limited set of simple interface crack problems. However, their utility for a wide range of interfacial crack problems is yet to be established.

Krueger, Ronald↗
NASA NTRS1984

Shape design sensitivity analysis of built-up structures

Selection of the best shape of a fillet in a tension bar such that no yielding occurs has long attracted the attention of engineers. Dimensions and notations for the bar and fillet are shown. With symmetry, only the upper half of the bar is considered. The optimal design problem is to find a boundary shape to minimize the total area of the fillet such that no yielding occurs. The classical boundary value problem is reduced to a variational or energy related problem which not only has excellent properties of existence and uniqueness but also provides the mathematical foundation for finite element analysis. The variational formulation may be viewed as the principle of virtual work and the finite element method as a application of the Galerkin method to the variational equation for approximate solution of the boundary value problem.

Choi, K. K.↗
NASA NTRS1981

Pull-out fibers from composite materials at high rate of loading

Numerical and experimental results are presented on the pullout phenomenon in composite materials at a high rate of loading. The finite element method was used, taking into account the existence of a virtual shear deformation layer as the interface between fiber and matrix. Experimental results agree well with those obtained by the finite element method. Numerical results show that the interlaminar shear stress is time dependent, in addition, it is shown to depend on the applied load time history. Under step pulse loading, the interlaminar shear stress fluctuates, finally decaying to its value under static loading.

Amijima, S.↗
DOE OSTI2024

Digital image correlation and infrared thermography data for seven unique geometries of 304L stainless steel

Material Testing 2.0 (MT2.0) is a paradigm that advocates for the use of rich, full-field data, such as from digital image correlation and infrared thermography, for material identification. By employing heterogeneous, multi-axial data in conjunction with sophisticated inverse calibration techniques such as finite element model updating and the virtual fields method, MT2.0 aims to reduce the number of specimens needed for material identification and to increase confidence in the calibration results. To support continued development, improvement, and validation of such inverse methods—specifically for rate-dependent, temperature-dependent, and anisotropic metal plasticity models—we provide here a thorough experimental data set for 304L stainless steel sheet metal. The data set includes full-field displacement, strain, and temperature data for seven unique specimen geometries tested at different strain rates and in different material orientations. Commensurate extensometer strain data from tensile dog bones is provided as well for comparison. We believe this complete data set will be a valuable contribution to the experimental and computational mechanics communities, supporting continued advances in material identification methods.

36 MATERIALS SCIENCE↗