Showing papers on "Smoothed finite element method published in 1996"

Error analysis and estimation for the finite volume method with applications to fluid flows

Abstract: The accuracy of numerical simulation algorithms is one of main concerns in modern Computational Fluid Dynamics. Development of new and more accurate mathematical models requires an insight into the problem of numerical errors. In order to construct an estimate of the solution error in Finite Volume calculations, it is first necessary to examine its sources. Discretisation errors can be divided into two groups: errors caused by the discretisation of the solution domain and equation discretisation errors. The first group includes insufficient mesh resolution, mesh skewness and non-orthogonality. In the case of the second order Finite Volume method, equation discretisation errors are represented through numerical diffusion. Numerical diffusion coefficients from the discretisation of the convection term and the temporal derivative are derived. In an attempt to reduce numerical diffusion from the convection term, a new stabilised and bounded second-order differencing scheme is proposed. Three new methods of error estimation are presented. The Direct Taylor Series Error estimate is based on the Taylor series truncation error analysis. It is set up to enable single-mesh single-run error estimation. The Moment Error estimate derives the solution error from the cell imbalance in higher moments of the solution. A suitable normalisation is used to estimate the error magnitude. The Residual Error estimate is based on the local inconsistency between face interpolation and volume integration. Extensions of the method to transient flows and the Local Residual Problem error estimate are also given. Finally, an automatic error-controlled adaptive mesh refinement algorithm is set up in order to automatically produce a solution of pre-determined accuracy. It uses mesh refinement and unrefinement to control the local error magnitude. The method is tested on several characteristic flow situations, ranging from incompressible to supersonic flows, for both steady-state and transient problems.

A finite element method for interface problems in domains with smooth boundaries and interfaces

Abstract: This paper is concerned with the analysis of a finite element method for nonhomogeneous second order elliptic interface problems on smooth domains. The method consists in approximating the domains by polygonal domains, transferring the boundary data in a natural way, and then applying a finite element method to the perturbed problem on the approximate polygonal domains. It is shown that the error in the finite element approximation is of optimal order for linear elements on a quasiuniform triangulation. As such the method is robust in the regularity of the data in the original problem.

Hierarchical bases and the finite element method

Abstract: In this work we present a brief introduction to hierarchical bases, and the important part they play in contemporary finite element calculations. In particular, we examine their role in a posteriori error estimation, and in the formulation of iterative methods for solving the large sparse sets of linear equations arising from finite element discretization.

Free mesh method: A new meshless finite element method

Abstract: A new meshless finite element method, named as the Free Mesh Method, is proposed in this paper. Once nodes are arranged in the domain to be analyzed, some temporary triangular elements are set around a node, i.e. a current central node. The contributions from the element matrices of the above temporary elements are assemebled to the total stiffness matrix. The above processes are performed on all the nodes in the domain. Finally, the solution is obtained by solving the total stiffness equation system as the usual finite element method. To demonstrate the effectiveness of the method, a simple two-dimensional heat conduction problem is solved.

Finite element solution of nonlinear intrinsic equations for curved composite beams

Abstract: A geometrically nonlinear finite element analysis, based on a weak form of the geometrically exact intrisic equilibrium and constitutive equations, is presented for initially curved and twisted composite beams. Results for both nonlinear static deformation and linearized free vibration about the static state of deformation are obtained and compared with published exact and theoretical analyses in the literature for initially curved isotropic beams and for isotropic and composite beams with swept tips. T h e results agree very well with experiment.

A finite element model for sound transmission through foam‐lined double‐panel structures

Abstract: In this paper, methods for coupling both elastic porous material (i.e., foam) and structural finite elements with either modal or finite element representations of acoustical system are presented. In addition, interface conditions are described for coupling elastic porous material finite elements with acoustical and structural finite elements in various configurations. The foam finite element is based on the elastic porous material theory of Biot. By considering sound transmission through layered systems placed in a waveguide, the accuracy of the coupled acoustical‐structural‐foam finite element model has been verified by comparing its transmission loss predictions with analytical solutions for the matching cases of infinite lateral extent. The constraint conditions at the edges of both the foam lining and the facing panels were found to have a significant effect on the normal incidence sound transmission loss of the double panel system at low frequencies.

Finite element analysis of orthogonal metal cutting mechanics

Abstract: The plane-strain finite element method is developed and applied to analyze the orthogonal metal cutting with continuous chip formation. Detailed work-material modeling, which includes the coupling of large strain, high strain-rate and temperature effects, is implemented. The versatility of the finite element method is demonstrated by presenting simulation results to complement the experimental measurements and to gain better understanding of the mechanics of the tool-chip contact and work-material deformation. The contour plots are used to show the distribution of parameters in the deformation zones. The finite difference method is applied to estimate the rate of change of parameters with respect to time. The Eulerian description of the deformation of work-material is also presented to show the variation at seven selected elements, which are expected to pass the deformation zones or go underneath the worn cutting tool.

Mixed finite element methods for flow in porous media

Abstract: Mixed nite element discretizations for problems arising in ow in porous medium applications are considered. We rst study second order elliptic equations which model single phase ow. We consider the recently introduced expanded mixed method. Combined with global mapping techniques, the method is suitable for full conductivity tensors and general geometry domains. In the case of the lowest order Raviart-Thomas spaces, quadrature rules reduce the method to cell-centered nite diierences, making it very eecient computationally. We consider problems with discontinuous coeecients on multiblock domains. To obtain accurate approximations, we enhance the scheme by introducing Lagrange multiplier pressures along subdomain boundaries and coeecient discontinuities. This modiication comes at no extra computational cost, if the method is implemented in parallel, using non-overlapping domain decomposition algorithms. Moreover, for regular solutions, it provides optimal convergence and discrete superconvergence for both pressure and velocity. We next consider the standard mixed nite element method on non-matching grids. We introduce mortar pressures along the non-matching interfaces. The mortar space is chosen to have higher approximability than the normal trace of the velocity spaces. The method is shown to be optimally convergent for all variables. Superconvergence for the subdomain pressures and, if the tensor coeecient is diagonal , for the velocities and the mortar pressures is also proven. We also consider the expanded mixed method on general geometry multiblock domains with non-matching grids. We analyze the resulting nite diierence scheme and show superconvergence for all variables. EEciency is not sacriiced by adding the mortar pressures. The computational complexity is shown to be comparable to the one on matching grids. Numerical results are presented, that verify the theory. iii We nally consider mixed nite element discretizations for the nonlinear multi-phase ow system. The system is reformulated as a pressure and a saturation equation. The methods described above are directly applied to the elliptic or parabolic pressure equation. We present an analysis of a mixed method on non-matching grids for the saturation equation of degenerate parabolic type. Acknowledgments To my wife Maria I would like to express my deep thanks to my advisor Mary Wheeler for her guidance and support. She has been both a mentor and a friend to me throughout my years at Rice. I also thank my former advisor Raytcho Lazarov, who is responsible for a great part of what I have achieved in my life. I am especially in debt to Todd Arbogast, the collaboration with …

Surgery Simulation Using fast Finite Elements

Abstract: This paper describes our recent work on real-time Surgery Simulation using Fast Finite Element models of linear elasticity [1]. In addition we discuss various improvements in terms of speed and realism.

An energy conserving co-rotational procedure for non-linear dynamics with finite elements

Abstract: A new procedure is proposed for implicit dynamic analysis using the finite element method. The main aim is to give stable solutions with large time-steps in the presence of significant rigid body motions, in particular rotations. In contrast to most conventional approaches, the time integration strategy is closely linked to the “element technology” with the latter involving a form of co-rotational procedure. For the undamped situation, the solution procedure leads to an algorithm that exactly conserves energy when constant external forces are applied (i.e. with gravity loading).

Circuit theoretical approach to couple two-dimensional finite element models with external circuit equations

Abstract: This paper presents a new method to couple two-dimensional finite element models with circuit equations. The method is based on handling of the finite element model as a circuit theoretical multiport element. This multiport element is treated in the same way as ordinary nonlinear circuit elements within the Newton-Raphson iteration of the circuit equations. The method has been utilized in a simulator program for analyzing power electronic drives in the time domain. The electrical machine is modeled by the two-dimensional finite element method. The power electronic circuit and the connections of the windings of the machine may have an arbitrary topology which is given by a net-list file (SPICE-type input file). The applicability of the method is investigated with two example cases which are verified by measurements. According to tests, the method proposed provides an effective and reliable way to construct simulators including finite element modeling.

A finite element method for the compressible Stokes equations

Abstract: A linearized steady-state, compressible, viscous Navier–Stokes system is considered. A finite element method is formulated. The unique solvability and stability of the finite element solution follow from a theorem for an abstract formulation. It is proved that when the subspaces for velocity and pressure satisfy the inf-sup condition associated with the (incompressible) Stokes system, the finite element method is uniquely solvable. An error estimate is obtained for the numerical approximation.

THE p-VERSION OF THE FINITE ELEMENT METHOD IN INCREMENTAL ELASTO-PLASTIC ANALYSIS

Abstract: Whereas the higher-order versions of the finite elements method (the p- and hp-version) are fairly well established as highly efficient methods for monitoring and controlling the discretization error in linear problems, little has been done to exploit their benefits in elasto-plastic structural analysis. Aspects of incremental elasto-plastic finite element analysis which are particularly amenable to improvements by the p-version is discussed. These theoretical considerations are supported by several numerical experiments. First, an example for which an analytical solution is available is studied. It is demonstrated that the p-version performs very well even in cycles of elasto-plastic loading and unloading, not only as compared to the traditional h-version but also in respect to the exact solution. Finally, an example of considerable practical importance - the analysis of a cold-worked lug - is presented which demonstrates how the modeling tools offered by higher-order finite element techniques can contribute to an improved approximation of practical problems.

Refined Three-Dimensional Finite Element Model for End-Plate Connection

Abstract: A finite element methodology is employed in the investigation of the behavioral characteristics of the end-plate connections. In order for the actual behavior to be simulated correctly and effectively, a three-dimensional (3D) model is established with nonconforming 3D elements. The elastoplastic nonconforming solid elements with variable nodes are used for the effective and economic 3D finite element modeling and analysis. The effect of bolt pretensioning and the shapes of the bolt shank, head, and nut are taken into consideration in the modeling. A simple yet efficient contact algorithm with a new gap element is employed to simulate the interaction between the end plate and column flange. The prototype end-plate connection is analyzed with the refined 3D finite element model and is verified by comparison with results from one particular test.

Computing cavity modes using the p-version of the finite element method

Abstract: We compare, theoretically and numerically, a standard edge finite element scheme for computing resonant modes of a cavity with two p-version finite element methods. In the p-version of the finite element method, accuracy is obtained by using a high-order polynomial space. For the example problem considered here, we show that both p-version methods offer significant advantages over the h-version edge code.

Periodic structure analysis using a hybrid finite element method

Abstract: A numerical solution for the time harmonic electromagnetic fields in a unit cell of an infinite, planar periodic structure has been developed and validated. It uses a variation of the hybrid finite element method that includes periodic radiation conditions at the exterior surfaces and periodicity conditions at the unit cell walls inside the structure. This technique allows the method to deal with structures that include inhomogeneous dielectrics and conductors with arbitrary shape and orientation. This paper presents the formulation of the finite element problem and shows the results of test cases involving an inductive screen, an artificial dielectric, and a layered-dielectric bandgap medium. The results show that the new method is accurate and versatile.

Analysis and finite element methods for a fluid-solid interaction problem in one dimension

Abstract: In this paper we study a time-harmonic fluid-solid interaction model problem in one dimension. This is a Helmholtz-type system equipped with boundary and transmission conditions. We show the existence of a unique solution to this problem and study its stability and regularity properties. We analyze the convergence of finite element methods with respect to appropriate energy norms. Computational results are also presented.

An adaptive finite element method for turbulent heat transfer

Abstract: This paper presents an adaptive finite element method for solving incompressible turbulent flows, including heat transfer effects, using a k e model of turbulence. Solutions are obtained in primitive variable using a highly accurate quadratic finite element on unstructured grids. Turbulence is modeled using the k e model. A projection error estimator is presented that takes into account the relative importance of the errors in velocity, temperature, pressure and turbulence variables, including the eddy viscosity. The efficiency and convergence rate of the methodology are evaluated by solving problems with known analytical solutions. The method is then applied to heated jets, heat transfer in a channel and finally, to heat transfer over a backward facing step. In all cases predictions are compared to experiments. Nomenclature c r i' 1' 2' e E f h k K n P P Pe Pr u V, W, S, T P E T k 6 model constants error roughness parameter body force element size turbulent kinetic energy Karman constant outward unit vector pressure production term Peclet number Prandtl number velocity vector friction velocity test function temperature density turbulent dissipation stress tensor V Vn A. a 8 0 av h T strain rate tensor wall shear stress gradient operator divergence viscosity thermal conductivity boundary element size for new mesh Subscripts initial value average finite element solution turbulent Copyright © 1996 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. INTRODUCTION Many aerodynamic and industrial flows involve the solution of the energy equation coupled to momentum and continuity to determine either their thermodynamic state or their heat transfer characteristics. Adaptive finite element methods provide a powerful approach for tackling such complex computational fluid dynamics problems. They can provide accurate solutions at a reasonable cost by automatically clustering elements around flow features of interest such as 1 American Institute of Aeronautics and Astronautics shear and boundary layers and reattachment points. The adaptive process is also cost effective in the sense that the best numerical solution is obtained at the least computational cost. Moreover, such approaches provide flexibility in modeling and algorithm development. The ability of the methodology to produce uniformly accurate solutions makes it possible to obtain 'numerically exact solutions' (grid independent) to the equations of motion, so that mathematical models of the physical phenomenon of interest can be evaluated with some confidence. Initial breakthroughs in adaptive computation were achieved in aerodynamics because of the pressing need for accurate computations of shock waves '. However, little work has been done for incompressible flows and even less for turbulent flow problems. Proof of concept computations for laminar incompressible flow were reported in Ref. [2,3]. A paper discussing adaptivity and the k e is that of Ref. [4] where structured grid are adapted by both moving nodes and imbedding a finer grid in the coarse one. This approach has led to solution improvements. However, the authors performed only one pass of adaptation. The degree of solution improvement is thus limited by the structured nature of the mesh and the limited number of refinements that are easily be implemented (only one step of refinement was implemented). Reference [5] presents applications of an adaptive finite element method to turbulent compressible flows with shocks. Turbulence is modeled by a low Reynolds number &-e model. The adaptive remeshing procedure significantly improved the accuracy of the predictions. However, adaptation was driven by an error estimator that is only sensitive to velocity gradients. Large variations in the turbulence kinetic energy, its dissipation and the eddy viscosity were ignored. The methodology proposed here is free of such ethic limitations. The use of unstructured grids provides for very highly localized grid resolution at a reasonable cost. The remeshing procedure also makes it possible to achieve any preset level of accuracy. The method can thus be viewed as a technique for generating 'numerically exact (grid independent) solutions to the differential equations modeling turbulent flow and heat transfer. In Ref. [6-9] the methodology proposed by the authors was quantitatively validated by solving laminar flows with known analytical solution and by computing cases for which experimental measurements were available. The methodology was further extended to convective heat transfer flows with variable fluid properties 10 and to zero-equation and two-equation models of turbulence for free shear flows l . The authors recently demonstrated the applicability of the proposed methodology for the k e and k u> models of turbulence applied to internal isothermal flows. The current work extends the methodology to turbulent flows with heat transfer effects. The methodology is based on adaptive remeshing coupled to a finite element solver for steady-state incompressible turbulent flows for which turbulence is represented by the k-€ model. The proposed error estimation technique and adaptive methodology are applicable to a wider class of problems than is treated here. The approach is valid for aerodynamic flows, such as those treated by Vemaganti, and to internal flows such as those found in combustors. The paper is organized as follows: First we describe the modeling of the problem. The equations of motion and the finite element solver are reviewed. The turbulence model is discussed and details of the nonlinear equation solver and wall boundary conditions are presented. The methodology section describes the error estimator and the adaptive remeshing strategy. The proposed methodology is then validated by solving problems with known analytical solutions to clearly quantify the accuracy improvements due to adaptivity. The method is then applied to turbulent flow in a heated channel and over a heated backward facing step for which experimental data are available. The paper closes with conclusions. MODELING OF THE PROBLEM Reynolds-averaged Navier-Stokes equations The flow regime of interest is modeled by the Reynoldsaveraged Navier-Stokes and energy equations: American Institute of Aeronautics and Astronautics

A simple finite element model for cloth drape simulation

Abstract: Uses Newmark’s method to carry out a time‐stepping finite element analysis to predict the behaviour of a cloth garment as it falls from an initial horizontal position to a final position draped around a human body form. Bases the finite element model on a simple beam element, in order to minimize the computational time. Accounts for large displacement behaviour by including the element geometric stiffness. Bases the body form on anthropomorphic data produced by a shadow scanner. Enlists a novel scheme to model the contact between the cloth and the underlying body form. Uses the finite element model to provide data for an animated display and finds that it produces sufficiently realistic results for the garment designer’s purposes.