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• Variational formulation of high-performance finite elements: Parametrized variational principles

  C. Felippa, C. Militello

  Computers and Structures (1990). Vol. 36 (1), pp. 1-11

  
  

  Abstract

  High-performance (HP) elements are simple finite elements constructed to deliver engineering accuracy with coarse arbitrary grids. This paper is part of a series on the variational basis of HP elements, with emphasis on those constructed with the free formulation (FF) and assumed natural strain (ANS) methods. The present paper studies parametrized variational principles that provide a foundation for the FF and ANS methods, as well as for a combination of both methods.

  Abstract
  High-performance (HP) elements are simple finite elements constructed to deliver engineering accuracy with coarse arbitrary grids. This paper is part of a series on [...]

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• Acoustic fluid volume modeling by the displacement potential formulation, with emphasis on the wedge element

  C. Felippa, J. DeRuntz

  Computers and Structures (1991). Vol. 41 (4), pp. 669-686

  
  

  Abstract

  This paper studies three-dimensional finite elements for modeling acoustic fluid volumes using the displacement potential formulation. These elements find application in shock and vibration analysis of fluid-structure interaction problems such as submerged structures and flexible containers. The paper concentrates on the derivation and implementation of the six-node wedge element because of its unconventional nature. This element is validated on the Bleich-Sandler problem of fluid-structure transient analysis.

  Abstract
  This paper studies three-dimensional finite elements for modeling acoustic fluid volumes using the displacement potential formulation. These elements find application [...]

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• Membrane triangles with corner drilling freedoms—II. The ANDES element

  C. Felippa, C. Militello

  Finite Elements in Analysis and Desing (1992). Vol. 12 (3-4), pp. 189-201

  
  

  Abstract

  This is the second article in a three-part series on the construction of 3-node, 9-dof membrane elements with normal-to-its-plane rotational freedoms (the so-called drilling freedoms) using parametrized variational principles. In this part, one such element is derived within the context of the assumed natural deviatoric strain (ANDES) formulation. The higher-order strains are obtained by constructing three parallel-to-sides pure-bending modes from which natural strains are obtained at the corner points and interpolated over the element. To attain rank sufficiency, an additional higher-order “torsional” mode, corresponding to equal hierarchical rotations at each corner with all other motions precluded, is incorporated. The resulting formulation has five free parameters. When these parameters are optimized against pure bending by energy balance methods, the resulting element is found to coalesce with the optimal EFF element derived in Part I. Numerical integration as a strain filtering device is found to play a key role in this achievement.

  Abstract
  This is the second article in a three-part series on the construction of 3-node, 9-dof membrane elements with normal-to-its-plane rotational freedoms (the so-called [...]

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• Membrane triangles with corner drilling freedoms—I. The EFF element

  K. Alvin, H. Fuente, B. Haugen, C. Felippa

  Finite Elements in Analysis and Desing (1992). Vol. 12 (3-4), pp. 163-187

  
  

  Abstract

  This paper is the first of a three-part series that studies the formulation of 3-node, 9-dof membrane elements with normal-to-element-plane rotations (the so-called drilling freedoms) within the context of parametrized variational principles. These principles supply a unified basis for several advanced element-construction techniques; in particular: the free formulation (FF), the extended free formulation (EFF) and the assumed natural deviatoric strain (ANDES) formulation. In Part I we construct an element of this type using the EFF. This derivation illustrates the basic steps in the application of that formulation to the construction of high-performance, rank-sufficient, nonconforming elements with corner rotations. The element is initially given the twelve degress of freedom of the linear strain triangle (LST), which allows the displacement expansion to be a complete quadratic in each component. The expansion basis contains the six linear basic functions and six energy-orthogonal quadratic higher-order functions. Three degrees of freedom, defined as the midpoint deviations from linearity along the triangle-median directions, are eliminated by kinematic constraints. The remaining hierarchical midpoint freedoms are transformed to corner rotations. The performance of the resulting element is evaluated in Part III.

  Abstract
  This paper is the first of a three-part series that studies the formulation of 3-node, 9-dof membrane elements with normal-to-element-plane rotations (the so-called drilling [...]

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• Parametrized variational principles encompassing compressible and incompressible elasticity

  C. Felippa

  International Journal of Solids and Structures (1992). Vol. 29 (1), pp. 57-68

  
  

  Abstract

  A parametrized five-field variational principle that can accommodate both compressible and incompressible hyperelasticity is presented. The primary variables are mean and deviatoric stresses, mean and deviatoric strains, and displacements. Through appropriate selection of parameters the functional of this general principle specializes to those previously presented by Atluri-Reissner, Herrmann and Franca.

  Abstract
  A parametrized five-field variational principle that can accommodate both compressible and incompressible hyperelasticity is presented. The primary variables are mean [...]

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• Parametrized variational principles for micropolar elasticity

  C. Felippa

  International Journal of Solids and Structures (1992). Vol. 29 (22), pp. 2709-2721

  
  

  Abstract

  A parametrized six-field variational principle for micropolar compressible linear elasticity is presented. The primary variables are symmetric and skew stresses, symmetric and skew strains, micropolar rotations and displacements. The governing functional is characterized by six free parameters. The connection between this formulation and the functionals with relaxed stress symmetry and independent rotations fields proposed by Reissner and Hughes-Brezzi for classical (non-polar) linear elasticity is examined. It is shown that the Hughes-Brezzi functionals are special cases of the parametrized functional but that the Reissner functionals are not. The former may be interpreted as a regularization (consistent stabilization) of the Reissner functionals that places them within the framework of micropolar elasticity. An eight-field parametrized principle that accounts for couple stresses is briefly described in the Appendix.

  Abstract
  A parametrized six-field variational principle for micropolar compressible linear elasticity is presented. The primary variables are symmetric and skew stresses, symmetric [...]

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• Membrane triangles with corner drilling freedoms— III. Implementation and performance evaluation

  C. Felippa, S. Alexander

  Finite Elements in Analysis and Desing (1992). Vol. 12 (3-4), pp. 203-239

  
  

  Abstract

  This paper completes a three-part series on the formulation of 3-node, 9-dof membrane triangles with corner drilling freedoms based on parametrized variational principles. The first four sections cover element implementation details including determination of optimal parameters and treatment of distributed loads. Then three elements of this type, labeled ALL, FF and EFF-ANDES, are tested on standard plane stress problems. ALL represents numerically integrated versions of Allman's 1988 triangle; FF is based on the free formulation triangle presented by Bergan and Felippa in 1985; and EFF-ANDES represent two different formulations of the optimal triangle derived in Parts I and II. The numerical studies indicate that the ALL, FF and EFF-ANDES elements are comparable in accuracy for elements of unitary aspect ratios. The ALL elements are found to stiffen rapidly in inplane bending for high aspect ratios, whereas the FF and EFF elements maintain accuracy. The EFF and ANDES implementations have a moderate edge in formation speed over the FF.

  Abstract
  This paper completes a three-part series on the formulation of 3-node, 9-dof membrane triangles with corner drilling freedoms based on parametrized variational principles. [...]

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• Comparison of strain recovery techniques for the mixed iterative method

  J. Cervenka, S. Keating, C. Felippa

  Computer Methods in Applied Mechanics and Engineering (1993). Vol. 9 (11), pp. 925-932

  
  

  Abstract

  Three strain recovery techniques are investigated in the framework of the mixed iterative method. The need for appropriate strain recovery for low-order elements is shown. The three techniques compared are: C-lumping, strain-smoothing and a new technique called C-splitting. It is shown that the use of C-lumping has negative effects on the convergence of the mixed iterative method. Both strain-smoothing and C-splitting deliver convergent results, but the new technique converges faster.

  Abstract
  Three strain recovery techniques are investigated in the framework of the mixed iterative method. The need for appropriate strain recovery for low-order elements is shown. [...]

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• Three-dimensional non-linear Timoshenko beam based on the core-congruential formulation

  L. Crivelli, C. Felippa

  International Journal for Numerical Methods in Engineering (1993). Vol. 36 (21), pp. 3647-3673

  
  

  Abstract

  A three-dimensional, geometrically non-linear, two-node Timoshenko beam element based on the total Lagrangian description is derived. The element behaviour is assumed to be linear elastic, but no restrictions are placed on the magnitude of finite rotations. The resulting element has twelve degrees of freedom: six translational components and six rotational-vector components. The formulation uses the Green-Lagrange strains and second Piola-Kirchhoff stresses as energy-conjugate variables and accounts for bending-stretching and bending-torsional-coupling effects without special provisions. The core-congruential formulation (CCF) is used to derive the discrete equations in a staged manner. Core equations involving the internal force vector and tangent stiffness matrix are developed at the particle level. A sequence of matrix transformations carries these equations to beam cross-sections and finally to the element nodal degrees of freedom. The choice of finite rotation measure is made in the next-to-last transformation stage, and the choice of over-the-element interpolation in the last one. The tangent stiffness matrix is found to retain symmetry if the rotational vector is chosen to measure finite rotations. An extensive set of numerical examples are presented to test and validate the present element.

  Abstract
  A three-dimensional, geometrically non-linear, two-node Timoshenko beam element based on the total Lagrangian description is derived. The element behaviour is assumed to be [...]

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• A survey of parametrized variational principles and applications to computational mechanics

  C. Felippa

  Computer Methods in Applied Mechanics and Engineering (1994). Vol. 113 (1-2), pp. 109-139

  
  

  Abstract

  This survey paper describes recent developments in the area of parametrized variational principles (PVPs) and selected applications to finite-element computational mechanics. A PVP is a variational principle containing free parameters that have no effect on the Euler-Lagrange equations. The theory of single-field PVPs, based on gauge functions (also known as null Lagrangians) is a subset of the Inverse Problem of Variational Calculus that has limited value. On the other hand, multifield PVPs are more interesting from theoretical and practical standpoints. Following a tutorial introduction, the paper describes the recent construction of multifield PVPs in several areas of elasticity and electromagnetics. It then discusses three applications to finite-element computational mechanics: the derivation of high-performance finite elements, the development of element-level error indicators, and the construction of finite element templates. The paper concludes with an overview of open research areas.

  Abstract
  This survey paper describes recent developments in the area of parametrized variational principles (PVPs) and selected applications to finite-element computational mechanics. [...]

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