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NASA NTRS1993

A variational method for finite element stress recovery and error estimation

A variational method for obtaining smoothed stresses from a finite element derived nonsmooth stress field is presented. The method is based on minimizing a functional involving discrete least-squares error plus a penalty constraint that ensures smoothness of the stress field. An equivalent accuracy criterion is developed for the smoothing analysis which results in a C sup 1-continuous smoothed stress field possessing the same order of accuracy as that found at the superconvergent optimal stress points of the original finite element analysis. Application of the smoothing analysis to residual error estimation is also demonstrated.

Tessler, A.↗

NASA NTRS1990

Application of a finite element algorithm for high speed viscous flows using structured and unstructured meshes

A higher-order streamline upwinding Petrov-Galerkin finite element method is employed for high speed viscous flow analysis using structured and unstructured meshes. For a Mach 8.03 shock interference problem, successive mesh adaptation was performed using an adaptive remeshing method. Results from the finite element algorithm compare well with both experimental data and results from an upwind cell-centered method. Finite element results for a Mach 14.1 flow over a 24 degree compression corner compare well with experimental data and two other numerical algorithms for both structured and unstructured meshes.

Vemaganti, Gururaja R.↗

NASA NTRS1997

Dynamic Electrothermal Model of a Sputtered Thermopile Thermal Radiation Detector for Earth Radiation Budget Applications

The Clouds and the Earth's Radiant Energy System (CERES) is a program sponsored by the National Aeronautics and Space Administration (NASA) aimed at evaluating the global energy balance. Current scanning radiometers used for CERES consist of thin-film thermistor bolometers viewing the Earth through a Cassegrain telescope. The Thermal Radiation Group, a laboratory in the Department of Mechanical Engineering at Virginia Polytechnic Institute and State University, is currently studying a new sensor concept to replace the current bolometer: a thermopile thermal radiation detector. This next-generation detector would consist of a thermal sensor array made of thermocouple junction pairs, or thermopiles. The objective of the current research is to perform a thermal analysis of the thermopile. Numerical thermal models are particularly suited to solve problems for which temperature is the dominant mechanism of the operation of the device (through the thermoelectric effect), as well as for complex geometries composed of numerous different materials. Feasibility and design specifications are studied by developing a dynamic electrothermal model of the thermopile using the finite element method. A commercial finite element-modeling package, ALGOR, is used.

Weckmann, Stephanie↗

DOE OSTI2018

Design and convergence analysis of the conforming virtual element method for polyharmonic problems

In the recent years, there has been a tremendous interest to numerical methods that approximate partial differential equations (PDEs) on computational meshes with arbitrarily-shaped polygonal/polyhedral (polytopal, for short) elements. A nonexhaustive list of such methods include the Mimetic Finite Difference method [4, 9, 21, 24, 28, 33, 35, 62, 64–67], the Polygonal Finite Element Method [68, 73, 74], the polygonal Discontinuous Galerkin Finite Element Methods [5, 7, 8, 10, 13–15, 18, 36, 38, 39, 41, 50], the Hybridizable Discontinuous Galerkin and Hybrid High-Order Methods [46–48, 51], the Gradient Discretization method [49, 53, 54], the Finite Volume Method [52], and the BEM-based FEM [71]. An alternative approach that also proved to be very successful is provided by the Virtual Element method (VEM), which was originally proposed in [19] for the numerical treatment of second-order elliptic problems [43, 44], and readily extended to linear and nonlinear elasticity [22, 25, 56], plate bending problems [37], Cahn-Hilliard equation [3], Stokes equations [2], Laplace-Beltrami equation [55], Darcy-Brinkam equation [76], discrete topology optimization problems [6], fracture networks problems [29], eigenvalue problems [59, 75]. The mixed virtual element formulation was proposed in [20, 34]; the nonconforming Virtual element formulations was proposed for second-order elliptic problems in [16], and later extended to general advection-reaction-diffusion problems, Stokes equation, the biharmonic problems, the iegenvalue problem, and the Schrodinger equation in [11, 30, 42, 45, 58, 80]. The p- and hp-version of the VEM were developed in [26, 27] and efficient multigrid methods for the resulting linear system of equations in [12]. A posteriori error estimates can be found in [40].

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2004

Higher-Order Finite Elements for Computing Thermal Radiation

Two variants of the finite-element method have been developed for use in computational simulations of radiative transfers of heat among diffuse gray surfaces. Both variants involve the use of higher-order finite elements, across which temperatures and radiative quantities are assumed to vary according to certain approximations. In this and other applications, higher-order finite elements are used to increase (relative to classical finite elements, which are assumed to be isothermal) the accuracies of final numerical results without having to refine computational meshes excessively and thereby incur excessive computation times. One of the variants is termed the radiation sub-element (RSE) method, which, itself, is subject to a number of variations. This is the simplest and most straightforward approach to representation of spatially variable surface radiation. Any computer code that, heretofore, could model surface-to-surface radiation can incorporate the RSE method without major modifications. In the basic form of the RSE method, each finite element selected for use in computing radiative heat transfer is considered to be a parent element and is divided into sub-elements for the purpose of solving the surface-to-surface radiation-exchange problem. The sub-elements are then treated as classical finite elements; that is, they are assumed to be isothermal, and their view factors and absorbed heat fluxes are calculated accordingly. The heat fluxes absorbed by the sub-elements are then transferred back to the parent element to obtain a radiative heat flux that varies spatially across the parent element. Variants of the RSE method involve the use of polynomials to interpolate and/or extrapolate to approximate spatial variations of physical quantities. The other variant of the finite-element method is termed the integration method (IM). Unlike in the RSE methods, the parent finite elements are not subdivided into smaller elements, and neither isothermality nor other unrealistic physical conditions are assumed. Instead, the equations of radiative heat transfer are integrated numerically over the parent finite elements by use of a computationally efficient Gaussian integration scheme.

Gould, Dana C.↗

NASA NTRS1994

Comparison of Gap Elements and Contact Algorithm for 3D Contact Analysis of Spiral Bevel Gears

Three dimensional stress analysis of spiral bevel gears in mesh using the finite element method is presented. A finite element model is generated by solving equations that identify tooth surface coordinates. Contact is simulated by the automatic generation of nonpenetration constraints. This method is compared to a finite element contact analysis conducted with gap elements.

Bibel, G. D.↗

NASA NTRS1989

A finite element based method for solution of optimal control problems

A temporal finite element based on a mixed form of the Hamiltonian weak principle is presented for optimal control problems. The mixed form of this principle contains both states and costates as primary variables that are expanded in terms of elemental values and simple shape functions. Unlike other variational approaches to optimal control problems, however, time derivatives of the states and costates do not appear in the governing variational equation. Instead, the only quantities whose time derivatives appear therein are virtual states and virtual costates. Also noteworthy among characteristics of the finite element formulation is the fact that in the algebraic equations which contain costates, they appear linearly. Thus, the remaining equations can be solved iteratively without initial guesses for the costates; this reduces the size of the problem by about a factor of two. Numerical results are presented herein for an elementary trajectory optimization problem which show very good agreement with the exact solution along with excellent computational efficiency and self-starting capability. The goal is to evaluate the feasibility of this approach for real-time guidance applications. To this end, a simplified two-stage, four-state model for an advanced launch vehicle application is presented which is suitable for finite element solution.

Bless, Robert R.↗

DOE OSTI2018

Hermite analogs of the lowest order Raviart–Thomas mixed method for convection–diffusion equations

The Raviart–Thomas mixed finite-element method of the lowest order (Raviart and Thomas in mixed finite-element methods for second-order elliptic problems, Lecture Notes in mathematics, Springer, New York, 1977) commonly known as the RT{sub 0} method, is a well-established and popular numerical tool to solve diffusion-like problems providing flux continuity across inter-element boundaries. Douglas and Roberts extended the method to the case of more general second-order boundary-value problems including the convection–diffusion equations (cf. this journal Douglas in Comput Appl Math 1:91–103; 1982). The main drawback of these methods, however, is the poor representation of the primal variable by piecewise constant functions. The Hermite analog of the RT{sub 0} method for treating pure diffusion phenomena proposed in Ruas (J Comput Appl Math, 246:234–242; 2013) proved to be a valid alternative to attain higher order approximation of the primal variable while keeping intact the matrix structure and the quality of the discrete flux variable of the original RT{sub 0} method. Non-trivial extensions of this method are studied here that can be viewed as Hermite analogs of the two Douglas and Roberts’ versions of the RT{sub 0} method, to solve convection–diffusion equations. A detailed convergence study is carried out for one of the Hermite methods, and numerical results illustrate the performance of both of them, as compared to each other and to the corresponding mixed methods.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1984

On the solution of problems involving impact type loading

The response of beam and plate members to pulse and impact loading is investigated, and numerical methods are analyzed and compared. The equations of motion, the finite difference and finite element methods, the time integration/Runge-Kutta techniques, and material modeling are discussed in detail. It is found that both the finite difference and the finite element methods could be accurately employed to discretize the spatial variation in the displacements. Central differences or a fourth order Runge-Kutta algorithm could be used for the time integration. The total energy of the system would give the stability and accuracy of the solution. Results showed that the finite element method provided better efficiency in obtaining accurate solution than the finite difference method when a scalar processor is used. However, the finite difference method was more efficient on the vector processor. Therefore, the most efficient method of solution depends on the type of computer present for the analysis.

Moyer, E. T., Jr.↗

DOE OSTI2018

A non-conforming finite volume element method for the two-dimensional Navier–Stokes/Darcy system

We consider the discretization of the stationary Navier–Stokes/Darcy system in a two-dimensional domain by the non-conforming finite volume element method. We use the standard formulation of the Navier–Stokes/Darcy system in the primitive variables and take as approximation space the non-conforming P{sub 1} elements for velocity and piezometric head and piecewise constant elements for the hydrostatic pressure. We prove that the unique solution of the non-conforming finite volume element method converges to the true solution with optimal order for velocity and piezometric head in discrete H{sup 1} norm and for pressure in discrete L{sup 2} norm, respectively. Finally, some numerical experiments are presented to validate our theoretical results.

42 ENGINEERING↗

DOE OSTI2024

Meshfree Methods

Meshfree methods have undergone substantial development and have received much attention in the last two decades. This new family of numerical methods is designed to inherit the main advantages of the finite element method such as compact supports of shape functions and good approximation properties while, at the same time, overcome the main disadvantages of the finite element method caused by the mesh dependence. The meshfree methods share a common feature that no mesh is needed and shape functions are constructed from sets of points, thus eliminating the need for time consuming mesh generation. The most significant advantage of meshfree methods is the flexibility in customizing approximation functions for desired regularity and for capturing essential physics and features of the particular problems of interest. Adaptivity formulation and multiple-scale solution strategies also can be implemented with relative ease. It has become clear that the meshfree methods provide considerable advantages over the conventional finite element methods in solving problems involving moving discontinuities, evolving material interfaces, multiple-scale phenomena, large material distortion and structural deformation, and fracture and damage processes. This Chapter gives an overview of many classes of meshfree methods, with more detailed discussions on Smoothed Particle Hydrodynamics (SPH), the Reproducing Kernel Particle Method (RKPM), Peridynamics (PD), the Material Point Method (MPM), as well as their applications in various challenging engineering problems.2

Chen, Jiun-Shyan↗

NASA NTRS1985

Analysis of Spiral Bevel Gearing

Program generalized for automated finite-element-method model generation. Tedious task of finite-element-method (FEM) model preparation greatly reduced by development of preprocessing program for automated model generation. Utility of program, including ability to rapidly evaluate parametric changes, demonstrated for spiral, helical, and spiral bevel gearing. Postprocessing program also developed, aids in review and interpretation of calculated FEM stresses.

Drago, R. J.↗

NASA NTRS2000

High Order Discontinuous Gelerkin Methods for Convection Dominated Problems with Application to Aeroacoustics

This project is about the investigation of the development of the discontinuous Galerkin finite element methods, for general geometry and triangulations, for solving convection dominated problems, with applications to aeroacoustics. On the analysis side, we have studied the efficient and stable discontinuous Galerkin framework for small second derivative terms, for example in Navier-Stokes equations, and also for related equations such as the Hamilton-Jacobi equations. This is a truly local discontinuous formulation where derivatives are considered as new variables. On the applied side, we have implemented and tested the efficiency of different approaches numerically. Related issues in high order ENO and WENO finite difference methods and spectral methods have also been investigated. Jointly with Hu, we have presented a discontinuous Galerkin finite element method for solving the nonlinear Hamilton-Jacobi equations. This method is based on the RungeKutta discontinuous Galerkin finite element method for solving conservation laws. The method has the flexibility of treating complicated geometry by using arbitrary triangulation, can achieve high order accuracy with a local, compact stencil, and are suited for efficient parallel implementation. One and two dimensional numerical examples are given to illustrate the capability of the method. Jointly with Hu, we have constructed third and fourth order WENO schemes on two dimensional unstructured meshes (triangles) in the finite volume formulation. The third order schemes are based on a combination of linear polynomials with nonlinear weights, and the fourth order schemes are based on combination of quadratic polynomials with nonlinear weights. We have addressed several difficult issues associated with high order WENO schemes on unstructured mesh, including the choice of linear and nonlinear weights, what to do with negative weights, etc. Numerical examples are shown to demonstrate the accuracies and robustness of the methods for shock calculations. Jointly with P. Montarnal, we have used a recently developed energy relaxation theory by Coquel and Perthame and high order weighted essentially non-oscillatory (WENO) schemes to simulate the Euler equations of real gas. The main idea is an energy decomposition under the form epsilon = epsilon(sub 1) + epsilon(sub 2), where epsilon(sub 1) is associated with a simpler pressure law (gamma)-law in this paper) and the nonlinear deviation epsilon(sub 2) is convected with the flow. A relaxation process is performed for each time step to ensure that the original pressure law is satisfied. The necessary characteristic decomposition for the high order WENO schemes is performed on the characteristic fields based on the epsilon(sub l) gamma-law. The algorithm only calls for the original pressure law once per grid point per time step, without the need to compute its derivatives or any Riemann solvers. Both one and two dimensional numerical examples are shown to illustrate the effectiveness of this approach.

Shu, Chi-Wang↗

NASA NTRS1972

Large deflection elastic-plastic dynamic response of stiffened shells of revolution

The formulation and check out porblems for a computer code DYNAPLAS, which analyzes the large deflection elastic-plastic dynamic response of stiffened shells of revolution, are presented. The formulation for special discretization is by the finite element method with finite differences being used for the evaluation of the pseudo forces due to material and geometric nonlinearities. Time integration is by the Houbolt method. The stiffeners may be due to concentrated or distributed eccentric rings and spring supports at arbitrary angles around the circumference of the elements. Check out porblems include the comparison of solutions from DYNAPLAS with experimental and other computer solutions for rings, conical and cylindrical shells and a curved panel. A hypothetical submarine including stiffeners and missile tube is studied under a combination of hydrostatic and dynamically applied asymmetrical pressure loadings.

Stricklin, J. A.↗

NASA NTRS1974

Large deflection elastic-plastic dynamic response of stiffened shells of revolution

This paper presents the formulation and check-out problems for a computer code DYNAPLAS, which analyzes the large deflection elastic-plastic dynamic response of stiffened shells of revolution. The formulation for spacial discretization is by the finite element method with finite differences being used for the evaluation of the pseudo forces due to material and geometric nonlinearities. Time integration is by the Houbolt method or central differences. The stiffeners may be due to concentrated or distributed eccentric rings and spring supports at arbitrary angles around the circumference of the elements. Check-out problems include the comparison of solutions from DYNAPLAS with experimental and other computer solutions for rings and conical and cylindrical shells. A hypothetical submarine including stiffeners and missile tube is studied under a combination of hydrostatic and dynamically applied asymmetrical pressure loadings.

Stricklin, J. A.↗

NASA NTRS1991

A combined finite element-boundary element formulation for solution of axially symmetric bodies

A new method is presented for the computation of electromagnetic scattering from axially symmetric bodies. To allow the simulation of inhomogeneous cross sections, the method combines the finite element and boundary element techniques. Interior to a fictitious surface enclosing the scattering body, the finite element method is used which results in a sparce submatrix, whereas along the enclosure the Stratton-Chu integral equation is enforced. By choosing the fictitious enclosure to be a right circular cylinder, most of the resulting boundary integrals are convolutional and may therefore be evaluated via the FFT with which the system is iteratively solved. In view of the sparce matrix associated with the interior fields, this reduces the storage requirement of the entire system to O(N) making the method attractive for large scale computations. The details of the corresponding formulation and its numerical implementation are described.

Collins, Jeffrey D.↗

DOE OSTI2021

Powder spreading, densification, and part deformation in binder jetting additive manufacturing

Binder jetting additive manufacturing (AM) can print complex structures in economical and scalable manner. Binder jetting AM comprises of deposition and weak binding of particles, known as green part, at room temperature and subsequent binder removal and sintering densification at high temperatures. However, during the densification (i.e., sintering), the part significantly deforms due to volume shrinkage. The deformation during sintering is difficult to predict, which prevents the widespread application of this technology. In this research, powder spreading simulation using discrete element method (DEM) was performed first to capture local variations in powder bed configuration. Second, finite element method (FEM) with a phenomenological continuum constitutive model was used to predict part shrinkage during the sintering process. DEM simulation showed variations in packing density, particle segregation, formation of uneven powder bed surface, and shift in particle size distribution (PSD). The sintering simulation modeled part deformation with a reasonable accuracy of 3% for solid-state sintering and intermediate liquid phase sintering. A demonstration case with non-uniform initial packing density showed that inhomogeneous green part density and PSD should be accounted for prediction of part deformation in binder jetting AM.

36 MATERIALS SCIENCE↗

NASA NTRS2016

Mesh Convergence Requirements for Composite Damage Models

The ability of the finite element method to accurately represent the response of objects with intricate geometry and loading renders the finite element method as an extremely versatile analysis technique for structural analysis. Finite element analysis is routinely used in industry to calculate deflections, stress concentrations, natural frequencies, buckling loads, and much more. The method works by discretizing complex problems into smaller, simpler approximations that are valid over small uniform domains. For common analyses, the maximum size of the elements that can be used is often be determined by experience. However, to verify the quality of a solution, analyses with several levels of mesh refinement should be performed to ensure that the solution has converged. In recent years, the finite element method has been used to calculate the resistance of structures, and in particular that of composite structures. A number of techniques such as cohesive zone modeling, the virtual crack closure technique, and continuum damage modeling have emerged that can be used to predict cracking, delaminations, fiber failure, and other composite damage modes that lead to structural collapse. However, damage models present mesh refinement requirements that are not well understood. In this presentation, we examine different mesh refinement issues related to the representation of damage in composite materials. Damage process zone sizes and their corresponding mesh requirements will be discussed. The difficulties of modeling discontinuities and the associated need for regularization techniques will be illustrated, and some unexpected element size constraints will be presented. Finally, some of the difficulties in constructing models of composite structures capable of predicting transverse matrix cracking will be discussed. It will be shown that to predict the initiation and propagation of transverse matrix cracks, their density, and their saturation may require models that are significantly more refined than those that have been contemplated in the past.

Davila, Carlos G.↗