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• A survey of the core-congruential formulation for geometrically nonlinear TL finite elements

  C. Felippa, L. Crivelli, B. Haugen

  Archives of Computational Methods in Engineering (1994). Vol. 1 (1), pp. 1-48

  
  

  Abstract

  This article presents a survey of the Core-Congruential Formulation (CCF) for geometrically nonlinear mechanical finite elements based on the Total Lagrangian (TL) kinematic description. Although the key ideas behind the CCF can be traced back to Rajasekaran and Murray in 1973, it has not subsequently received serious attention. The CCF is distinguished by a two-phase development of the finite element stiffness equations. The initial phase develop equations for individual particles. These equations are expressed in terms of displacement gradients as degrees of freedom. The second phase involves congruential-type transformations that eventually binds the element particles of an individual element in terms of its node-displacement degrees of freedom. Two versions of the CCF, labeled Direct and Generalized, are distinguished. The Direct CCF (DCCF) is first described in general form and then applied to the derivation of geometrically nonlinear bar, and plane stress elements using the Green-Lagrange strain measure. The more complex Generalized CCF (GCCF) is described and applied to the derivation of 2D and 3D Timoshenko beam elements. Several advantages of the CCF, notably the physically clean separation of material and geometric stiffnesses, and its independence with respect to the ultimate choice of shape functions and element degrees of freedom, are noted. Application examples involving very large motions solved with the 3D beam element display the range of applicability of this formulation, which transcends the kinematic limitations commonly attributed to the TL description.

  Abstract
  This article presents a survey of the Core-Congruential Formulation (CCF) for geometrically nonlinear mechanical finite elements based on the Total Lagrangian (TL) kinematic [...]

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• Superconducting axisymmetric finite elements based on a gauged potential variational principle—II. Solution and numerical results

  J. Schuler, C. Felippa

  Computing Systems in Engineering (1994). Vol. 5 (3), pp. 227-237

  
  

  Abstract

  Part II of a two part paper discusses an incremental-iterative nonlinear solution technique for solving the nonlinear finite element equations of the superconducting state of a superconductor. The untreated equations are highly ill-conditioned and are impossible to solve within the typical 16-place double precision supplied by most computers. A combination of matrix scaling and mesh grading techniques is used to reduce the condition number of the tangent stiffness matrix and increase the accuracy of the current carrying boundary layer representation. Numerical results for a one-dimensional model of a time-independent superconductor treated by the Ginzburg-Landau model are presented and discussed. The computed solutions clearly display the Meissner effect of magnetic field expulsion from the central region of the superconductor. These results are compared to the physics of a low-viscosity fluid problem. From this analogy, a physical argument is advanced about the macroscopic behavior of superconductors.

  Abstract
  Part II of a two part paper discusses an incremental-iterative nonlinear solution technique for solving the nonlinear finite element equations of the superconducting state [...]

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• Superconducting axisymmetric finite elements based on a gauged potential variational principle—I. Formulation

  J. Schuler, C. Felippa

  Computing Systems in Engineering (1994). Vol. 5 (3), pp. 215-225

  
  

  Abstract

  The present work is part of a research program for the numerical simulation of electromagnetic (EM) fields within conventional Ginzburg-Landau (GL) superconductors. The final goal of this research is to formulate, develop and validate finite element (FE) models that can accurately capture electromagnetic, thermal and material phase changes in a superconductor. The formulations presented here are for a time-independent Ginzburg-Landau superconductor and are derived from a potential-based variational principle. In Part I of this paper, we develop an appropriate variational formulation of time-independent superconductivity for the general three-dimensional case and specialize it to the one-dimensional case. Also developed are expressions for the material-dependent parameters α and β of GL theory and their dependence upon the temperature T. The one-dimensional formulation is then discretized for finite element purposes and the first variation of these equations is obtained. The resultant Euler equations contain nonlinear terms in the primary variables. To solve these equations, an incremental-iterative solution method is used. Expressions for the internal force vector, external force vector, loading vector and tangent stiffness matrix are therefore developed for use with the solution procedure.

  Abstract
  The present work is part of a research program for the numerical simulation of electromagnetic (EM) fields within conventional Ginzburg-Landau (GL) superconductors. The final [...]

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• 50 Year classic reprint: An appreciation of R. Courant's ‘variational methods for the solution of problems of equilibrium and vibrations,’ 1943

  C. Felippa

  International Journal for Numerical Methods in Engineering (1994). Vol. 37 (13), pp. 2159-2187

  
  

  Abstract

  Fifty years elapsed since the appearance of the title paper, in Bull. Am. Math. Soc., 49, 1-23 (1943), and twenty six since my original encounter with it. I first read it in 1967 during an internal workshop on the origins of the finite element method (FEM), held at UC Berkeley, where I was a Post-Doctoral Research Associate. I must admit at not having been overly impressed - the pride of having just finished a thesis in refined elements coupled with the arrogance of youth to dismiss what to me seemed like tricial hand calculations for a simple torsion problem. Why, a 1967 CDC 6600 computer wrap those up in milliseconds...

  Abstract
  Fifty years elapsed since the appearance of the title paper, in Bull. Am. Math. Soc., 49, 1-23 (1943), and twenty six since my original encounter with it. I first read it [...]

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• From the individual element test to finite element templates: Evolution of the patch test

  C. Felippa, B. Haugen, C. Militello

  International Journal for Numerical Methods in Engineering (1995). Vol. 38 (2), pp. 199-229

  
  

  Abstract

  This paper starts a sequence of three articles that follow an unconventional approach in finite element research. The ultimate objective is to construct high-performance elements and element-level error estimators for those elements. The approach takes off from our previous work in high-performance elements and culminates with the development of finite element templates. The present paper concentrates on the patch test and evolved versions of the test that have played a key role in this research. Following a brief review of the historical roots, we present the Individual Element Test (IET) of Bergan and Hanssen in an expanded context that encompasses several important classes of new elements. The relationship of the IET to the multielement forms A, B and C of the patch test and to the single-element test are investigated. An important consequence of the IET application is that the element stiffness equations decompose naturally into basic and higher-order parts. The application of this decomposition to the “sanitization” of the non-convergent BCIZ element is described and verified with numerical experiments. Two sequel papers in preparation are subtitled ‘the algebraic approach’ and ‘element-level error estimation’. These apply the fundamental decomposition to the derivation of templates for specific mechanical elements and to the construction of element-level error estimators, respectively.

  Abstract
  This paper starts a sequence of three articles that follow an unconventional approach in finite element research. The ultimate objective is to construct high-performance elements [...]

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• Parametric unification of matrix structural analysis: classical formulation and d-connected mixed elements

  C. Felippa

  Finite Element in Analysis and Design (1995). Vol. 21 (1-2), pp. 45-74

  
  

  Abstract

  Concepts and techniques from the field of Parametrized Variational Principles (PVPs) are extended to Matrix Structural Analysis (MSA). Free parameters are used as weighting factors of governing discrete equations. Combining this idea with matrix manipulation techniques yields a continuous spectrum of supermatrix equations. Setting parameters to numerical values provides specific solution methods, some of which are well known whereas others are not. The approach is applied to the classical MSA of truss and framework structures as well as to displacement-connected FE models generated by a parametrized mixed functional. The main advantage of this “top down” derivation of solution schemes is the unification of seemingly disjoint methods for instructional and classification purposes. In addition, the question of duality between range-space and null-space representations is clarified.

  Abstract
  Concepts and techniques from the field of Parametrized Variational Principles (PVPs) are extended to Matrix Structural Analysis (MSA). Free parameters are used as weighting [...]

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• Recent developments in parametrized variational principles for mechanics

  C. Felippa

  Computational Mechanics (1996). Vol. 18 (3), pp. 159-174

  
  

  Abstract

  This paper gives an introduction to the formulation of parametrized variational principles (PVPs) in mechanics. This is complemented by more advanced material describing selected recent developments in hybrid and nonlinear variational principles. A PVP is a variational principle containing free parameters that have no effect on the Euler-Lagrange equations and natural boundary conditions. The theory of single-field PVPs, based on gauge functions, is a subset of the Inverse Problem of Variational Calculus that has limited value. On the other hand, multified PVPs are more interesting from both theoretical and practical standpoints. The two-dimensional Poisson equation is used to present, in a tutorial fashion, the formulation of parametrized mixed functionals. This treatment is then extended to internal interfaces, which are useful in treatment of discontinuities, subdomain linkage and construction of parametrized hybrid functionals. This is followed by a similar but more compact treatment of three-dimensional classical elasticity, and a parametrization of nonlinear hyperelasticity.

  Abstract
  This paper gives an introduction to the formulation of parametrized variational principles (PVPs) in mechanics. This is complemented by more advanced material describing selected [...]

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• Parametrized variational principles in dynamics applied to the optimization of dynamic models of plates

  F. Castro, C. Miitello, C. Felippa

  Computational Mechanics (1997). Vol. 20 (4), pp. 285-294

  
  

  Abstract

  We investigate the use of Parametrized Variational Principles (PVP) in linear structural dynamics. Our main objective is to assess whether the free parameters can be used to enhance the accuracy of dynamic models on a fixed mesh. Consistent, boundary-consistent and lumped mass matrices are defined within the framework of the PVP. The accuracy provided by three different mass matrices in the computation of plate frequencies is numerically studied. A method to update the free parameters on an element by element basis to improve the dynamic model is presented. Numerical experiments that characterize that improvement for the vibration and transient-response analysis of plates are presented. These experiments suggest that such update is primarily beneficial for modes in the intermediate frequency range.

  Abstract
  We investigate the use of Parametrized Variational Principles (PVP) in linear structural dynamics. Our main objective is to assess whether the free parameters can be used [...]

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• Everyone may benefit from subsidising entry to risky occupations

  C. Felippa, K. Park, J. Black, D. Meza

  Journal of Public Economics (1997). Vol. 66 (3), pp. 409-424

  
  

  Abstract

  This paper shows that in the presence of costly state verification, directly or indirectly subsidising entry to risky occupations may benefit everyone. The result holds even in the presence of private insurance. Indeed, it may be desirable to prohibit private insurance in favour of subsidies to hazardous activities. These findings do not depend on the government having an advantage over the private sector in observing outcomes. The explanation is that through its influence on equilibrium price, feasible fiscal policy can shift the return distribution so as to create collective insurance more cheaply than is possible through private contracting with its requirement of costly auditing. Amongst applications is a case for a loss-making state bank offering high interest-rate loans.

  Abstract
  This paper shows that in the presence of costly state verification, directly or indirectly subsidising entry to risky occupations may benefit everyone. The result holds even [...]

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• An algebraically partitioned FETI method for parallel structural analysis: performance evaluation

  M. Justino, K. Park, C. Felippa

  International Journal for Numerical Methods in Engineering (1997). Vol. 40 (15), pp. 2739-2758

  
  

  Abstract

  This paper presents the algorithmic performance of an algebraically partitioned Finite Element Tearing and Interconnection (FETI) method presented in a companion paper. A simple structural assembly topology is employed to illustrate the implementation steps in a Matlab software environment. Numerical results indicate that the method is scalable, provided the iterative solution preconditioner employs the reduced interface Dirichlet preconditioner. A limited comparison of the present method with the differentially partitioned FETI method with corner modes is also offered. Based on this comparison and a reasonable extrapolation, we conclude the present algebraically partitioned FETI method possesses a similar iteration convergence property of the differentially partitioned FETI method with corner modes

  Abstract
  This paper presents the algorithmic performance of an algebraically partitioned Finite Element Tearing and Interconnection (FETI) method presented in a companion paper. A [...]

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