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This paper presents a new algorithm for solving systems of nonlinear equations, which combines elements of the method of local variations with a sequential secant method. The new algorithm converges superlinearly and its convergence does not depend upon a good initial guess at a solution.
Four Galerkin finite-element methods are tested for solving the free-boundary problem that describes steady solidification. The formulations differ in the solution method used to account for the unknown shape of the melt/solid interface, in the interphase condition (either balance of heat flux or equilibrium of temperature) distinguished for locating the interface, and in the technique used for solving the systems of algebraic equations that result from the finite-element approximations. Methods that use the melting point isotherm to locate the melt/solid interface are found more accurate and efficient than formulations based on the interfacial energy balance. Solution by a Galerkin-Newton algorithm of the free-boundary problem transformed to a fixed domain is most efficient when the field problem in each phase is made nonlinear by including radiation from the melt and solid to the surroundings.
The equations Laplacian operator omega = 0, (1.1a) and omega = Laplacian operator Chi, (1.1b) describe, in suitable units, 2-D Stokes flow of an incompressible fluid occupying a domain D in which omega is the vorticity and Chi is the stream function. The flow is uniquely determined by specifying the velocity on the boundary B of D, a condition which leads to specifying the stream function Chi and its normal derivative Chi sub n on B. A mathematically similar problem arises in describing the equilibrium of a flat plate in structural mechanics where a related 1-D problem by finite difference or finite element methods is to introduce effective methods for imposing the boundary conditions through which (1.1a) is coupled to (1.1b). These models thus provide a simple starting point for examining the general treatment of boundary conditions for more general time dependent Navier-Stokes incompressible flows. For the purpose of discussion it is assumed that D is a square domain. A standard finite difference method to solve (1.1) is to introduce a uniform grid and then use standard five point finite difference operators to express each equation in (1.1). At any point on the boundary B a value of Chi is specified by the boundary conditions but a value of omega at the same boundary mesh point will also be required to complete the computation. Methods are discussed which overcome the difficulty in solving these problems.
A combination of ab initio, atomistic and finite element methods (FEM) were used to investigate the structures, energetics and lattice thermal conductance of grain boundaries for the ultra high temperature ceramic ZrB2. Atomic models of idealized boundaries were relaxed using density functional theory. Information about bonding across the interfaces was determined from the electron localization function. The Kapitza conductance of larger scale versions of the boundary models were computed using non-equilibrium molecular dynamics. The interfacial thermal parameters together with single crystal thermal conductivities were used as parameters in microstructural computations. FEM meshes were constructed on top of microstructural images. From these computations, the effective thermal conductivity of the polycrystalline structure was determined.
The work investigates the applicability of the unresolved Computational Fluid Dynamics and Discrete Element Method (CFDDEM) technique based on empirical equations for fluid-particle coupling. We first carry out a series of representative volume element simulations using the high-resolution particle-resolved Lattice Boltzmann method and Discrete Element Method (LBMDEM) coupled by an Immersed Moving Boundary (IMB) scheme. Then, we compare the results obtained by both LBMDEM and empirical equations used in unresolved CFDDEM with analytical solutions. It is found that the existing empirical equations used in solving fluid-particle interactions in 2D CFDDEM fail to accurately calculate the hydrodynamic force applied to solid particles. The underlying reason is that the existing empirical models are obtained based on 3D experimental results and thus are not applicable to 2D problems. Based on the simulation results, a new drag coefficient model is then proposed. The estimated drag forces using the new model are compared favourably with the simulated ones, indicating the good performance of the proposed model.
Methods for the efficient computational prediction of the performance of phononic pseudocrystals (structured materials capable of blocking extraordinary ranges of frequency) in COMSOL and other comparable finite element method codes are set forth. These methods include boundary conditions that make possible halving the size of the computational domain. Also included is an introduction of elastic energy density methods for assessing the extinction of elastic waves within the patterned region.
The purpose of this paper is to show that most existing methods for the calculation of fluid flows can be interpreted as special applications of a single mathematical formalism. This formalism, called the generalized Galerkin method, then provides a single conceptual framework for comparing various methods in terms of convergence and accuracy. The method is presented in sufficient mathematical detail to permit its interpretation as a projection in function space. In order to demonstrate the basic thesis, finite difference, finite element, strip integral, and classical integral methods of boundary layer theory are developed by application of the method.
A method is developed for determining the acoustic admittance of a test liner installed in the wall of a grazing flow impedance tube. The mean flow is permitted flow gradients in both cross-sectional directions of the tube. The unknown admittance value is obtained by solving an eigenvalue problem. This eigenvalue problem results from the application of the finite-element method to the partial differential equation and boundary conditions governing the acoustic field. The credibility of the method is established by comparing results with exact solutions obtained for a constant mean-flow profile and with previous results for cases involving shear in only one cross-sectional direction. Excellent comparisons are obtained in both cases. The analysis is used in conjunction with a limited amount of experimental data and shows that the flow must be accurately modeled in order to determine the acoustic-liner properties.
Crack propagation in a rotating inner raceway of a high speed roller bearing is analyzed using the boundary integral equation method. The method consists of an edge crack in a plate under tension, upon which varying Hertzian stress fields are superimposed. A computer program for the boundary integral equation method was written using quadratic elements to determine the stress and displacement fields for discrete roller positions. Mode I and Mode II stress intensity factors and crack extension forces G sub 00 (energy release rate due to tensile opening mode) and G sub r0 (energy release rate due to shear displacement mode) were computed. These calculations permit determination of that crack growth angle for which the change in the crack extension forces is maximum. The crack driving force was found to be the alternating mixed-mode loading that occurs with each passage of the most heavily loaded roller. The crack is predicted to propagate in a step-like fashion alternating between radial and inclined segments, and this pattern was observed experimentally. The maximum changes DeltaG sub 00 and DeltaG sub r0 of the crack extension forces are found to be good measures of the crack propagation rate and direction.
A unified computed aided design method was studied for the cure cycle design that incorporates an optimal design technique with the analytical model of a composite cure process. The preliminary results of using this proposed method for optimal cure cycle design are reported and discussed. The cure process of interest is the compression molding of a polyester which is described by a diffusion reaction system. The finite element method is employed to convert the initial boundary value problem into a set of first order differential equations which are solved simultaneously by the DE program. The equations for thermal design sensitivities are derived by using the direct differentiation method and are solved by the DE program. A recursive quadratic programming algorithm with an active set strategy called a linearization method is used to optimally design the cure cycle, subjected to the given design performance requirements. The difficulty of casting the cure cycle design process into a proper mathematical form is recognized. Various optimal design problems are formulated to address theses aspects. The optimal solutions of these formulations are compared and discussed.
Alloy 718 crack growth experiments were conducted to assess the ability of the selected path-independent (P-I) integrals to describe the elevated temperature crack growth behavior. These tests were performed on single edge notch (SEN) specimens under displacement control with multiple extensometers to monitor the specimen and crack mouth opening displacement (CMOD). The displacements in these tests were sufficiently high to induce bulk cyclic inelastic deformation of the specimen. Under these conditions, the linear elastic fracture mechanics (LEFM) parameter K does not correlate the crack growth data. The experimentally measured displacement gradients at the end of specimen gage length were used as the boundary conditions in elastic-plastic finite element method (FEM) analyses. These analyses were performed with a node release approach using CYANIDE, a GEAE FEM code, which included a gap element which is capable of efficiently simulating crack closure. Excellent correlation was obtained between the experimentally measured and predicted variation of stress and CMOD with crack length and the stress-CMOD loops for Alloy 718 tests conducted at 538 C. This confirmed the accuracy of the FEM crack growth simulation approach. The experimentally measured crack growth rate data correlated well the selected P-I integrals. These investigations have produced significant progress in developing P-I integrals as non-linear fracture mechanics parameters. The results suggest that this methodology has the potential of accurately describing elevated temperature crack growth behavior under the combined influence of thermal cycling and bulk elastic-inelastic deformation states.
A low-dimensional dynamic model for transitional buoyancy-driven flow in a differentially heated tall enclosure is presented. The full governing partial differential equations with the associated boundary conditions are solved by a spectral element method for a cavity of aspect ratio A=20. Proper orthogonal decomposition is applied to the oscillatory solution at Prandtl number Pr=P tau (omega) = 0.71 and Grashof number G tau (omega) = 3.2 x 10 (exp 4) to construct empirical eigenfunctions. Using the four most energetic empirical eigenfunctions for the velocity and temperature as basis functions and applying Galerkin's method, a reduced model consisting of eight nonlinear ordinary differential equations is obtained. Close to the 'design' conditions (P tau(omega) G tau(omega)), the low-order model (LOM) predictions are in excellent agreement with the predictions of the full model. In particular, the critical Grashof number at the onset of the first temporal flow instability (Hopf bifurcation) was well as the frequency and amplitude of oscillations at supercritical conditions are in excellent agreement with the predictions of the full model. Far from the 'design' conditions, the LOM predicts the existence of multiple stable steady solutions at large values of G tau, and a unique stable steady solution at small values of G tau, and exhibits hysteretic behavior that is qualitatively similar to that observed in direct numerical simulations based on the full model.
We analyze the convergence of a substructuring iterative method with Lagrange multipliers, proposed recently by Farhat and Roux. The method decomposes finite element discretization of an elliptic boundary value problem into Neumann problems on the subdomains and a coarse problem for the subdomain nullspace components. For linear conforming elements and preconditioning by the Dirichlet problems on the subdomains, we prove the asymptotic bound on the condition number C(1 + log(H/h))(sup gamma), gamma = 2 or 3, where h is the characteristic element size and H is the subdomain size.
Numerical simulations are used to examine the roughness-induced transient growth in a laminar boundary-layer flow. Based on the spectral element method, these simulations model the stationary disturbance field associated with a nonsmooth roughness geometry, such as the spanwise periodic array of circular disks used by White and co-workers during a series of wind tunnel experiments at Case Western Reserve University. Besides capturing the major trends from the recent measurements by White and Ergin, the simulations provide additional information concerning the relative accuracy of the experimental findings derived from two separate wall-finding procedures. The paper also explores the dependence of transient growth on geometric characteristics of the roughness distribution, including the height and planform shape of the roughness element and the ratio of roughness due to spacing between an adjacent pair of elements. Results are used for a preliminary assessment of the differences between recently reported theoretical results of Tumin and Reshotko and the measurements by White and Ergin.
A localized 3-D roughness causes boundary-layer separation and (weak) shocks. Most importantly, streamwise vortices occur which induce streamwise (low U, high T) streaks. Immersed boundary method (volume force) suitable to represent roughness element in DNS. Favorable comparison between bi-global stability theory and DNS for a "y-mode" Outlook: Understand the flow physics (investigate "z-modes" in DNS through sinuous spanwise forcing, study origin of the beat in DNS).
An immersed boundary method for the compressible Navier-Stokes equation and the additional infrastructure that is needed to solve moving boundary problems and fully coupled fluid-structure interaction is described. All the methods described in this paper were implemented in NASA's LAVA solver framework. The underlying immersed boundary method is based on the locally stabilized immersed boundary method that was previously introduced by the authors. In the present paper this method is extended to account for all aspects that are involved for fluid structure interaction simulations, such as fast geometry queries and stencil computations, the treatment of freshly cleared cells, and the coupling of the computational fluid dynamics solver with a linear structural finite element method. The current approach is validated for moving boundary problems with prescribed body motion and fully coupled fluid structure interaction problems in 2D and 3D. As part of the validation procedure, results from the second AIAA aeroelastic prediction workshop are also presented. The current paper is regarded as a proof of concept study, while more advanced methods for fluid structure interaction are currently being investigated, such as geometric and material nonlinearities, and advanced coupling approaches.
The origin of spurious solutions in computational electromagnetics, which violate the divergence equations, is deeply rooted in a misconception about the first-order Maxwell's equations and in an incorrect derivation and use of the curl-curl equations. The divergence equations must be always included in the first-order Maxwell's equations to maintain the ellipticity of the system in the space domain and to guarantee the uniqueness of the solution and/or the accuracy of the numerical solutions. The div-curl method and the least-squares method provide rigorous derivation of the equivalent second-order Maxwell's equations and their boundary conditions. The node-based least-squares finite element method (LSFEM) is recommended for solving the first-order full Maxwell equations directly. Examples of the numerical solutions by LSFEM for time-harmonic problems are given to demonstrate that the LSFEM is free of spurious solutions.
Two iterative schemes based on the mixed finite element method are developed for analyzing steady natural convection in a melt adjacent to its solid phase. The simplest method decouples the calculation of the field variables and the shape of the melt/solid interface into two interlocked iterations that are performed successively. The second method uses Newton's iteration to solve simultaneously for both types of unknowns and has a quadratic convergence rate. Results for a model problem of melt and solid in a cylindrical ampoule show the Newton algorithm to be a factor of three more efficient.