Search NASA⌕ Search

SEARCH · Search NASA

Results for “finite element problems”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 145 records · Page 8
All sourcesNASA NTRSDOE OSTI
DOE OSTI2019

A simple design of highly birefringent and nonlinear photonic crystal fiber with ultra-flattened dispersion

We present a compressed hexagonal photonic crystal fiber (CH-PCF) based on a simple structure with ultra-flattened dispersion, high birefringence and nonlinearity. The propagation characteristics are studied accurately by utilizing full vector finite-element method (FEM). Numerical results indicate that the birefringence of 1.59 × 10{sup −2} was obtained with high nonlinear coefficient of 42.58 W{sup −1} km{sup −1}(X) and 52.80 W{sup −1} km{sup −1}(Y) at an effective wavelength of 1.55 μm. Furthermore, CH-PCF presents an ultra-flattened dispersion with variation of ± 3.4 ps/nm/km(X) and ± 1.6 ps/nm/km(Y) over 1.3 μm to 1.8 μm (0.5 μm bandwidth), respectively. In addition, this circular-hole structure can be fabricated conveniently at most easy compared with complex designs, which is an advantage for practical applications such as dispersion compensation and nonlinear optics application.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

Determination of Thermoelastic Properties of Carbon Nanotube/Epoxy Composites Using Finite Element Method

In the present study, three-dimensional finite element (FE) analysis has been performed for determination of thermoelastic properties of carbon nanotube (CNT)/epoxy composites considering both linear and nonlinear stress–strain relations of CNT. A square representative volume element has been considered for the FE analysis (FEA), and the effect of considering nonlinear stress–strain relation of CNT on the prediction of thermoelastic properties has been determined by comparing the results with the theoretical results. Results show that in order to obtain realistic values of thermoelastic properties, the nonlinear stress–strain relation of CNT needs to be used in the FEA. Not considering nonlinear stress–strain relation of CNTs in the FEA may lead to estimating erroneous values of thermoelastic properties of CNT composites which may have adverse effect in the design of structural components made of such composites subjected to thermomechanical loading.

36 MATERIALS SCIENCE↗
DOE OSTI2019

Nonlinear Creep Deformation of Polycarbonate at High Stress Level: Experimental Investigation and Finite Element Modeling

It is well known that polycarbonate (PC) undergoes time-dependent deformation (i.e., creep deformation), and nonlinear creep deformation is often experienced at high stress level. Using the time–temperature–stress superposition principle (TTSSP), we obtain a new master curve, which covers higher stress level, and successfully establish a new modeling method of creep deformation of PC. First, to investigate the effect of applied stress level on the creep compliance (i.e., stress-dependent nonlinear creep deformation), this study conducted various creep tests with eight different stress levels. We found that the creep compliance curve strongly depended on the applied stress level; in particular, a higher stress level induced a larger difference in creep compliance. According to the TTSSP, the creep compliance curve at each stress level shifts with the creep time (i.e., stress reduced time). When we appropriately selected the stress reduced time, we obtained the master curve of creep compliance, which is unified with respect to various applied stresses. However, we found that the stress-shifted factor is not compliant with the previous TTSSP, especially in the higher stress regime. Therefore, this regime was also considered to obtain a new master curve that can cover a wide range of stress levels. Finally, our established creep model (master curve and stress shift factor) was introduced into FEM, and then this numerical model was verified by comparison with experimental data. Our model may be useful for predicting the creep deformation of PC subjected to a wide range of applied stresses.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Enhanced multiscale restriction-smoothed basis (MsRSB) preconditioning with applications to porous media flow and geomechanics

A novel method to enable application of the Multiscale Restricted Smoothed Basis (MsRSB) method to non M-matrices is presented. The original MsRSB method is enhanced with a filtering strategy enforcing M-matrix properties to enable the robust application of MsRSB as a preconditioner. Through applications to porous media flow and linear elastic geomechanics, the method is proven to be effective for scalar and vector problems with multipoint finite volume (FV) and finite element (FE) discretization schemes, respectively. Realistic complex (un)structured two- and three-dimensional test cases are considered to illustrate the method's performance.

58 GEOSCIENCES↗
DOE OSTI2020

Advanced Newton methods for geodynamical models of Stokes flow with viscoplastic rheologies

Strain localization and resulting plasticity and failure play an important role in the evolution of the lithosphere. These phenomena are commonly modeled by Stokes flows with viscoplastic rheologies. The nonlinearities of these rheologies make the numerical solution of the resulting systems challenging, and iterative methods often converge slowly or not at all. Yet accurate solutions are critical for representing the physics. Moreover, for some rheology laws, aspects of solvability are still unknown. We study a basic but representative viscoplastic rheology law. The law involves a yield stress that is independent of the dynamic pressure, referred to as von Mises yield criterion. Two commonly used variants, perfect/ideal and composite viscoplasticity, are compared. We derive both variants from energy minimization principles, and we use this perspective to argue when solutions are unique. We propose a new stress-velocity Newton solution algorithm that treats the stress as an independent variable during the Newton linearization but requires solution only of Stokes systems that are of the usual velocity-pressure form. To study different solution algorithms, we implement 2-D and 3-D finite element discretizations, and we generate Stokes problems with up to 7 orders of magnitude viscosity contrasts, in which compression or tension results in significant nonlinear localization effects. Comparing the performance of the proposed Newton method with the standard Newton method and the Picard fixed-point method, we observe a significant reduction in the number of iterations and improved stability with respect to problem nonlinearity, mesh refinement, and the polynomial order of the discretization.

Rudi, Johann↗
DOE OSTI2023

Numerical Study of Coupled Water and Vapor Flow, Heat Transfer, and Solute Transport in Variably-Saturated Deformable Soil During Freeze-Thaw Cycles

As climate change intensifies, soil water flow, heat transfer, and solute transport in the active, unfrozen zones within permafrost and seasonally frozen ground exhibit progressively more complex interactions that are difficult to elucidate with measurements alone. For example, frozen conditions impede water flow and solute transport in soil, while heat and mass transfer are significantly affected by high thermal inertia generated from water-ice phase change during the freeze-thaw cycle. To assist in understanding these subsurface processes, the current study presents a coupled two-dimensional model, which examines heat conduction-convection with water-ice phase change, soil water (liquid water and vapor) and groundwater flow, advective-dispersive solute transport with sorption, and soil deformation (frost heave and thaw settlement) in variably saturated soils subjected to freeze-thaw actions. This coupled multiphysics problem is numerically solved using the finite element method. The model's performance is first verified by comparison to a well-documented freezing test on unsaturated soil in a laboratory environment obtained from the literature. Then based on the proposed model, we quantify the impacts of freeze-thaw cycles on the distribution of temperature, water content, displacement history, and solute concentration in three distinct soil types, including sand, silt and clay textures. The influence of fluctuations in the air temperature, groundwater level, hydraulic conductivity, and solute transport parameters was also comparatively studied. The results show that (a) there is a significant bidirectional exchange between groundwater in the saturated zone and soil water in the vadose zone during freeze-thaw periods, and its magnitude increases with the combined influence of higher hydraulic conductivity and higher capillarity; (b) the rapid dewatering ahead of the freezing front causes local volume shrinkage within the non-frozen region when the freezing front propagates downward during the freezing stage and this volume shrinkage reduces the impact of frost heave due to ice formation. This gradually recovers when the thawed water replenishes the water loss zone during the thawing stage; and (c) the profiles of soil moisture, temperature, displacement, and solute concentration during freeze-thaw cycles are sensitive to the changes in amplitude and freeze-thaw period of the sinusoidal varying air temperature near the ground surface, hydraulic conductivity of soil texture, and the initial groundwater levels. Our modeling framework and simulation results highlight the need to account for coupled thermal-hydraulic-mechanical-chemical behaviors to better understand soil water and groundwater dynamics during freeze-thaw cycles and further help explain the observed changes in water cycles and landscape evolution in cold regions.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2019

A Fast Solver for the Fractional Helmholtz Equation

The purpose of this paper is to study a Helmholtz problem with a spectral fractional Laplacian, instead ofthe standard Laplacian. Recently, it has been established that such a fractional Helmholtz problem better captures the underlying behavior in Geophysical Electromagnetics. We establish the well-posedness and regularity of this problem. We introduce a hybrid finite element-spectral approach to discretize it and show well-posedness of the discrete system. In addition, we derive a priori discretization error estimates. Finally, we introduce an efficient solver that scales as well as the best possible solver for the classical integer-order Helmholtz equation. We conclude with several illustrative examples that confirm our theoretical findings.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Lumping Techniques for Mixed Finite Element Diffusion Discretizations

We are interested in positivity preserving spatial discretizations applicable to arbitrary order mixed finite element spatial discretizations of the radiative diffusion equations. Though not guaranteed to yield strict positivity, matrix lumping is appealing as a method of increasing solution positivity because unlike ad hoc fixups or strictly non-negative, non-linear solution representations, matrix lumping keeps the mixed finite element diffusion equations as a linear system of equations. Self-lumping schemes have previously been shown to be a generalization of more traditional matrix lumping techniques. Self-lumping schemes restrict integration quadrature points to the interpolation points used to define interpolatory finite element basis functions. As shown previously, the positivity and order of accuracy of self-lumping schemes are directly related to the choice of finite element interpolation point. Here, we develop self-lumping schemes for arbitrary order, mixed finite element spatial discretizations of the diffusion equation in Cartesian geometry. After introducing several possible self-lumping schemes appropriate for mixed finite elements, we test the ability of each scheme to increase solution positivity relative to an unlumped discretization using a simple test problem designed to induce negative solutions, evaluating efficacy as a function of finite element order and finite element interpolation point type. The method of manufactured solutions is then used to evaluate order of convergence for promising self-lumping schemes.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2018

Superconvergence analysis for nonlinear parabolic equation with EQ{sub 1}{sup rot} nonconforming finite element

EQ{sub 1}{sup rot} nonconforming finite element method (FEM) applied to a class of nonlinear parabolic equation is discussed. First, by use of two typical characters of this element (one is that the associated FE interpolation operator is identical to its traditional Ritz projection operator; the other is that the consistency error is of order O(h{sup 2}), one order higher than the interpolation error, when the exact solution of the problems belongs to H{sup 3}(Ω)), the supercloseness of order O(h{sup 2}) in broken H{sup 1}-norm for semi-discrete scheme is obtained without the boundedness of numerical solution in L{sup ∞}-norm through splitting the numerical solution into several parts. Moreover, we get the desired result with requirement of u,u{sub t}∈H{sup 3}(Ω) only. Secondly, a linearized Crank–Nicolson fully discrete scheme is proposed and the superclose property of order O(h{sup 2}+τ{sup 2}) is derived by constructing a suitable auxiliary problem. Finally, a numerical example is carried out to confirm our theoretical analysis. Here, h is the subdivision parameter and τ is the time step.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

A Higher Order, Stable Partitioned Scheme for Fluid-Structure Interaction Problems

Although still a very active area of research with many open questions, there exist highly-accurate and efficient algorithms for numerically estimating solutions to the equations of fluid motion. Similarly, great strides have been made in numerically modeling the motion of solids. However, there exist many practical applications where one must solve both fluid and structure in tandem. There is a gap in the literature for high order unconditionally stable partitioned algorithms for problems of this type. We investigate a new partitioned scheme which shows promise in filling this gap. We use a finite-element-based model with an added mass approach on problems coupling the Euler beam equation with the incompressible Navier-Stokes equations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

A note on the reliability of goal-oriented error estimates for Galerkin finite element methods with nonlinear functionals

Here, we consider estimating the discretization error in a nonlinear functional J (u) in the setting of an abstract variational problem: find u ϵ $\mathscr{V}$ such that B (u, φ) = L (φ) ∀φ ϵ $\mathscr{V}$, as approximated by a Galerkin finite element method. Here, $\mathscr{V}$ is a Hilbert space, B (. , .) is a bilinear form, and L (∙) is a linear functional. We consider well-known error estimates η of the form J (u) - J (u h ) ≈ η = L (z) - B (u h , z), where u h denotes a finite element approximation to u, and z denotes the solution to an auxiliary adjoint variational problem. We show that there exist nonlinear functionals for which error estimates of this form are not reliable, even in the presence of an exact adjoint solution z. An estimate η is said to be reliable if there exists a constant C ϵ $\mathbb{R}$ >0 independent of u h such that |J (u) - J (u h )| ≤ C|η|. We present several example pairs of bilinear forms and nonlinear functionals where reliability of η is not achieved.

A posteriori↗
DOE OSTI2024

A layered solid finite element formulation with interlaminar enhanced displacements for the modeling of laminated composite structures

Accurate modeling of layered composite structures often requires the use of detailed finite element models which can sufficiently resolve the kinematics and material behavior within each layer of the composite. However, individually discretizing each material layer into finite elements presents a prohibitive computational expensive given the large number of thin layers comprising some laminated composites. To address these challenges, an 8-node layered solid hexahedral finite element is formulated with the aim of striking an appropriate balance between efficiency and fidelity. The element is discretized into an arbitrary number of distinct material layers, and employs reduced in-plane integration within each layer. The chosen reduced integration scheme is supplemented by a novel physical stabilization approach which includes layerwise enhancements to mitigate various forms of locking phenomena. The proposed framework additionally supports the inclusion of interlaminar enhanced displacements to better represent the kinematics of general layered composite materials. Finally, the described element formulation has been implemented in the ParaDyn finite element code, and its efficacy for modeling laminated composite structures is demonstrated on a variety of verification problems.

42 ENGINEERING↗
DOE OSTI2023

A Novel Partitioned Approach for Reduced Order Model—Finite Element Model (ROM-FEM) and ROM-ROM Coupling

Partitioned methods allow one to build a simulation capability for coupled problems by reusing existing single-component codes. In so doing, partitioned methods can shorten code development and validation times for multiphysics and multiscale applications. In this work, we consider a scenario in which one or more of the “codes” being coupled are projection-based reduced order models (ROMs), introduced to lower the computational cost associated with a particular component. We simulate this scenario by considering a model interface problem that is discretized independently on two non-overlapping subdomains. Here we then formulate a partitioned scheme for this problem that allows the coupling between a ROM “code” for one of the subdomains with a finite element model (FEM) or ROM “code” for the other subdomain. The ROM “codes” are constructed by performing proper orthogonal decomposition (POD) on a snapshot ensemble to obtain a low-dimensional reduced order basis, followed by a Galerkin projection onto this basis. The ROM and/or FEM “codes” on each subdomain are then coupled using a Lagrange multiplier representing the interface flux. To partition the resulting monolithic problem, we first eliminate the flux through a dual Schur complement. Application of an explicit time integration scheme to the transformed monolithic problem decouples the subdomain equations, allowing their independent solution for the next time step. We show numerical results that demonstrate the proposed method’s efficacy in achieving both ROM-FEM and ROM-ROM coupling.

de Castro, Amy↗
DOE OSTI2021

Scale-bridging with the extended/generalized finite element method for linear elastodynamics

This paper presents an extended/generalized finite element method for bridging scales in linear elastodynamics in the absence of scale separation. More precisely, the GFEMgl framework is expanded to enable the numerical solution of multiscale problems through the automated construction of specially-tailored shape functions, thereby enabling high-fidelity finite element modeling on simple, fixed finite element meshes. Furthermore, this introduces time-dependencies in the shape functions in that they are subject to continuous adaptation with time. The temporal aspects of the formulation are investigated by considering the Newmark-β time integration scheme, and the efficacy of mass lumping strategies is explored in an explicit time-stepping scheme. This method is demonstrated on representative wave propagation examples as well as a dynamic fracture problem to assess its accuracy and flexibility.

36 MATERIALS SCIENCE↗
DOE OSTI2018

Variational multiscale element-free Galerkin method combined with the moving Kriging interpolation for solving some partial differential equations with discontinuous solutions

Solving partial differential equations with discontinuous solutions is an important challenging problem in numerical analysis. To this end, there are some methods such as finite volume method, discontinuous Galerkin approach and particle technique that are able to solve these problems. In the current paper, the moving Kriging element-free Galerkin method has been combined with the variational multiscale algorithm to obtain acceptable and high-resolution solutions. For testing this technique, we select some PDEs with discontinuous solution such as Burgers’, Sod’s shock tube, advection–reaction–diffusion, Kuramoto–Sivashinsky, Boussinesq and shallow water equations. First, we obtain a time-discrete scheme by approximating time derivative via finite difference technique. Then we introduce the moving Kriging interpolation and also obtain their shape functions. We use the element-free Galerkin method for approximating the spatial derivatives. This method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the classical element-free Galerkin method test and trial functions are moving least squares approximation (MLS) shape functions. Since the shape functions of moving least squares (MLS) approximation do not have Kronecker delta property, we cannot implement the essential boundary condition, directly. Thus, we employ the shape functions of moving Kriging interpolation and radial point interpolation technique which have the mentioned property. Also, in the element-free Galekin method, we do not use any triangular, quadrangular or other type of meshes. The element-free Galerkin method is a global method while finite element method is a local one. This technique employs a background mesh for integration which makes it different from the truly mesh procedures. Several test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed schemes.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

Strong and Weak Scaling of the Sierra/SD Eigenvector Problem to a Billion Degrees of Freedom

Sierra/SD is a structural dynamics finite element software package that is known for its scalability and performance on DOE supercomputers. While there are historical documents demonstrating weak and strong scaling on DOE systems such as Redsky, no such formal studies have been done on modern architectures. This report demonstrates that Sierra/SD still scales on modern architectures. Non structured meshes in the shape of an I-Beam are solved in sizes ranging from fifty thousand degrees of freedom in serial up to one and a half billion degrees of freedom on over eighteen thousand processors using only default solver options. The report serves as a baseline for users to estimate computation cost of finite element analyses in Sierra/SD, understand how solver options relate to computational costs, and pick optimal processor counts to solve a given problem size, as well as a baseline for evaluating computational cost and scalability on next generation architectures.

42 ENGINEERING↗
DOE OSTI2021

Automatically adaptive stabilized finite elements and continuation analysis for compaction banding in geomaterials

Under compressive creep, viscoplastic solids experiencing internal mass transfer processes can accommodate singular cnoidal wave solutions as material instabilities at the stationary wave limit. These instabilities appear when the loading rate is significantly faster than the material's capacity to diffusive internal perturbations, leading to localized failure features (e.g., cracks and compaction bands). These cnoidal waves, generally found in fluids, have strong nonlinearities that produce periodic patterns. Due to the singular nature of the solutions, the applicability of the theory is currently limited. Additionally, practical simulation tools require proper regularization to overcome the challenges that singularity induces. We focus on the numerical treatment of the governing equation using a nonlinear approach building on a recent adaptive stabilized finite element method. This automatic refinement method provides an error estimate that drives mesh adaptivity, a crucial feature for the problem at hand. We compare the performance of this adaptive strategy against analytical and standard finite element solutions. We then investigate the sensitivity of the diffusivity ratio, the parameter controlling the process, and identify multiple possible solutions with several stress peaks. We also show the evolution of the spacing between peaks for all solutions as a function of that parameter.

42 ENGINEERING↗
DOE OSTI2022

Precise 3D reactor core calculation using spherical harmonics and discontinuous Galerkin finite element methods

We study the use of P{sub N} method in angle and discontinuous Galerkin is space to solve 3D neutron transport problem. P{sub N} method consists in developing the angular flux on truncated spherical harmonics basic. In this paper, we couple this method with the discontinuous finite elements in space to obtain a complete discretization of the multigroup neutron transport equation. To investigate its precision, the method was applied to Takeda and C5G7 benchmark problems. These calculations point out that the proposed P{sub N}-DG method is capable of producing accurate solutions in small computational time, and that it is able to handle complex 3D geometries. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗