Abstract: This note was written in response to questions concerning satisfaction of equilibrium conditions by finite element solutions computed by means of StressCheck. These questions were raised because StressCheck does not report nodal forces, however users experienced with conventional finite element analysis codes are expecting to see evidence of equilibrium based on summation of nodal forces. It is shown that (a) the nodal forces associated with each element are in equilibrium in both the h- and p-versions and (b) satisfaction of the equilibrium of nodal forces is a property of the displacement formulation and is unrelated to the quality or reliability of the finite element solution. The PowerPoint document illustrates the main points of the discussion with an example.
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The effects of residual tensile stresses induced by cold-working a fastener hole
Brot A. and Matias C., Israel Aerospace Industries, ASIP Conference (2007).
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Finite Element Analysis
Szabó B. and Babuška I., John Wiley and Sons Inc., New York (1991).
The p-version of the finite element method
Szabó B., Düster A. and Rank E., Encyclopedia of Computational Mechanics, Vol 1, Chapter 5, John Wiley and Sons Inc., New York (2004).
On the importance and uses of feedback information in FEA
Szabó B., Actis R., Applied Numerical Mathematics, Vol 52, pp. 219-234 (2005).
Procedures for the verification and validation of working models for structural shells
Szabó B., Muntges D., Journal of Applied Mechanics, Vol 72, pp. 907-915 (2005).
Quasi-regional mapping for the p-version of the finite element method
Királyfalvi G. and Szabó B., Finite Elements in Analysis and Design, Vol 27, pp. 85-97 (1997).
Hierarchic models for laminated plates and shells
Actis R., Szabó B. and Schwab C., Computational Methods in Applied Mechanics and Engineering, Vol 172, pp. 79-107 (1999).
Adaptive selection of polynomial degrees on a finite element mesh
Bertóti, E. and Szabó B., Int. J. Numer. Methods in Engineering, Vol 42, pp. 561-578 (1998).
Linear models of buckling and stress stiffening
Szabó B. and Királyfalvi G., Computational Methods in Applied Mechanics and Engineering, Vol 171, pp. 43-59 (1999).
Solution of contact problems using hp-version of the finite element method
Páczelt I., Szabó B. and Szabó T., Computers and Mathematics with Applications, Vol 38, pp. 49-69 (1999).
Superconvergent computations of flux intensity factors and first derivatives by the FEM
Szabó B. and Yosibash Z., Computational Methods in Applied Mechanics and Engineering, Vol 129, pp. 349-370 (1996).
A note on numerically computed eigenfunctions and generalized stress intensity factors associated with singular points
Yosibash Z. and Szabó B., Engineering Fracture Mechanics, Vol 54, pp. 593-595 (1996).
The p-version of the finite element method compared to an adaptive h-version for the deformation theory of plasticity
Düster A. and Rank E., Computational Methods in Applied Mechanics and Engineering, Vol 190, Number 15, pp. 1925-1935, January 2001.
Abstract: A p-version of the finite element method is applied to the deformation theory of plasticity and the results are compared to a state-of-the-art adaptive h-version. It is demonstrated that even for nonlinear elliptic problems the p-version is a very efficient discretization strategy.
A p-version finite element approach for two- and three-dimensional problems of the J2 flow theory with nonlinear isotropic hardening
Düster A. and Rank E., International Journal for Numerical Methods in Engineering, Vol 53, Issue 1, pp. 49-63, October 2001.
Abstract: In this paper an implementation of a two- and three-dimensional p-version approach to the J2 flow theory with non-linear isotropic hardening for small displacements and small strains is presented. Based on higher-order quadrilateral and hexahedral element formulations, a Newton-Raphson iteration scheme combined with a radial return algorithm is applied to find approximate solutions for the underlying physically non-linear model problem. Curved boundaries are taken care of with the blending function method, allowing an accurate representation of geometry with only a few p-elements. Numerical examples demonstrate, that the p-version supplies efficient and accurate approximations to this class of physically non-linear problems.
Nonlinear limit state analysis of laminated plates by p-version of F.E.M.
Woo K.S., Hong C.H., Lee C.G., Basu P.K.,
Abstract: A p-version finite element model based on degenerate shell element is proposed for the analysis of orthotropic laminated plates. In the nonlinear formulation of the model, the total Lagrangian formulation is adopted with large deflections and moderate rotations. The material model is based on the Huber-Mises yield criterion and Prandtl-Reuss flow rule in accordance with the theory of strain hardening yield function. The model is also based on extension of equivalent-single layer laminate theory (ESL theory) with shear deformation leading to continuous shear strains at the interface of two layers. The validity of the proposed p-version finite element model is demonstrated through several comparative points of view in terms of ultimate load, convergence characteristics, nonlinear effect, and shape of plastic zone.
High order solid elements for thin-walled structures with applications to linear and nonlinear structural analysis
Rank E., Düster A., Muthler A., Romberg R., European Congress on Computational Methods in Applied Sciences and Engineering, July 2004
Abstract: As an alternative to the well-known and widely used dimensionally reduced formulations for an approximation of thin-walled structures we will investigate in this paper the feasibility of strictly three-dimensional models, using high order elements being coupled to a precise geometric description of the structure. As a key issue, the p-version of the FEM is used, offering a consistent and accurate way to implement solid elements having large aspect ratio (up to a few hundred) and to represent much more general shapes of element surfaces than those available in the usual isoparametric approach. A transition from thin- to thick-walled constructions is thus possible without the necessity to couple models of differing dimensions and without imposing any restrictions on the (three-dimensional) kinematics of the structure. We will demonstrate our results in several numerical examples ranging from a strictly three-dimensional simulation of a building to spring-back analysis in metal forming processes.
P-FEM for a class of pressure/density dependent plasticity models with application to cold isostatic pressing
Frage N., Hartmann S., Holzer S., Rank E., Yosibash Z., Dariel M., German-Israeli Foundation for Scientific Research and Development, June 2006
Abstract: During the past decade, the p-version of the finite element method (FEM) was proven to be a reliable tool for simulating linear problems with many advantages compared to the classical h-version FEM. At the same time, an increasing interest has been observed for computing mechanical processes taking into account more realistic material behavior in geometrically and physically nonlinear situations for which classical h-FEMs may fail.
Control of geometry induced error in hp finite element (FE) simulations
Xue D., Demkowicz L., International Journal of Numerical Analysis and Modeling, Vol 2, Number 3, pp. 283-300, March 2005
Abstract: This paper discusses a general framework for handling curvilinear geometries in high accuracy finite element (FE) simulations, for both elliptic and maxwell problems. Based on the differential manifold concept, the domain is represented as a union of geometrical blocks prescribed with globally compatible, explicit or implicit parameterizations. The idea of parametric H - H(curl) - and H(div) - conforming elements is reviewed, and the concepts of exact geometry elements and isoparametric elements are discussed. Presented numerical examples indicate the necessity of accounting for the geometry error in FE error calculations, especially for the H(curl) problems.
Some aspects of coupling structural models and p-version finite element methods
Rank E., Düster A., Krafczyk M., Rucker M., 1998
Abstract: This paper addresses some questions arising from an integration of coupling structural models and p-version finite element methods. After a brief introduction to the p-version for Reissner-Mindlin plate problems we will consider the modelling of loads or elastic foundations acting only on parts of elements. Furthermore, we will address the question of a posteriori control of the accuracy of the approximation being important for reliable computations. The last part of the paper compares the computational efficiency of the p-version to low order approximations. Finally, it will be motivated why the use of the p-version is expected to be superior in parallel efficiency compared to standard h-version codes.