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• Customizing high performance elements by fourier methods

  C. Felippa

  (2001). Trends in computational structural mechanics.

  
  

  Abstract

  This paper illustrates, through a worked out example, the application of finite element templates to construct high performance bending elements for vibration and buckling problems. The example focuses on the improvement of the mass and geometric stiffness matrices of plane beams. Similar methods can be used for Kirchhoff plate bending elements, but this material is omitted because of space constraints. The process interweaves classical tools: Fourier analysis and orthogonal polynomials, with newer ones: finite element templates and computer algebra systems. Templates are parametrized algebraic forms that uniquely characterize an element population by a "genetic signature" defined by the set of free parameters. Specific elements are obtained by assigning numeric values to the parameters. This freedom of choice can be used to design custom elements. For the beam example, Legendre polynomials are used to construct templates for the material stiffness, geometric stiffness and mass matrices. Fourier analysis carried out through symbolic computation searches for template signatures of mass and stiffness that deliver matrices with desirable attributes for specific target situations. Three objectives are noted: high accuracy for vibration and buckling analysis, wide separation of acoustic and optical branches, and low sensitivity to mesh distortion and boundary conditions. This paper examines in some detail only the first objective

  Abstract
  This paper illustrates, through a worked out example, the application of finite element templates to construct high performance bending elements for vibration and buckling [...]

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• Interfacing nonmatching FEM meshes: the zero moment rule

  K. Park, C. Felippa, G. Rebel

  (2001). Trends in computational structural mechanics, pp. 1-13

  
  

  Abstract

  This is a tutorial paper that outlines, through a benchmark example, an efiective method for coupling nonmatched flnite element meshes. Coupling is done through a displacement frame interposed between the interface meshes. That frame is treated with a FEM discretization and \glued" to the meshes through localized Lagrange multipliers collocated at mesh-interface nodes. The approach can be used to couple meshes of arbitrary geometry, discretization type (e.g., FEM and BEM) and even meshes of difierent physics (e.g., structure and ∞uids). The example, however, focuses on a simplifled 2D case that can be explained within space constraints. The requirement for preservation of constant stress states leads to an easily visualized condition, called the zero-moment rule or ZMR, that can be used to locate frame nodes on frame geometry. The ZMR provides all possible consistent frame-node conflgurations. Generalizations of the 2D ZMR rule to 3D problems involve partitioning of the frame and are not reported here. Thus the ZMR is necessary but not su-cient. The constant stress consistency condition may be interpreted, according to the reader's taste, as either an interface patch test, a vanishing of the flrst variation of the interface potential under admissible kinematic modes, or a requirement for energy conservation across the interface.

  Abstract
  This is a tutorial paper that outlines, through a benchmark example, an efiective method for coupling nonmatched flnite element meshes. Coupling is done through a displacement [...]

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• Special issue: Advances in computational methods for fluid-structure interaction and coupled problems - Preface

  R. Ohayon, C. Felippa

  Computer Methods in Applied Mechanics and Engineering (2001). Vol. 190 (24-25), pp. 2977-2978

  
  

  Abstract

   The computer based simulation of Fluid-Structure Interaction (FSI) problems has historical and practical importance in computational mechanics. FSI was one of the first systematically developed areas in the wider framework of what is now called computational multiphysics. Its development propelled original advances in modeling, solution algorithms and computer implementation of coupled models coming from two different disciplines: solid and fluid mechanics. Despite that relatively early start, the practical importance of computational FSI has remained undiminished. Among its important scientific and technological applications we can cite the following.

  Abstract
   The computer based simulation of Fluid-Structure Interaction (FSI) problems has historical and practical importance in computational mechanics. FSI was one of the first [...]

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• Customizing the mass and geometric stiffness of plane beam elements by Fourier methods

  C. Felippa

  Engineering Computations (2001). Vol. 18 (1-2), pp. 286-303

  
  

  Abstract

  Teaches by example the application of finite element templates in constructing high performance elements. The example discusses the improvement of the mass and geometric stiffness matrices of a Bernoulli-Euler plane beam. This process interweaves classical techniques (Fourier analysis and weighted orthogonal polynomials) with newer tools (finite element templates and computer algebra systems). Templates are parameterized algebraic forms that uniquely characterize an element population by a "genetic signature" defined by the set of free parameters. Specific elements are obtained by assigning numeric values to the parameters. This freedom of choice can be used to design "custom" elements. For this example weighted orthogonal polynomials are used to construct templates for the beam material stiffness, geometric stiffness and mass matrices. Fourier analysis carried out through symbolic computation searches for template signatures of mass and geometric stiffness that deliver matrices with desirable properties when used in conjunction with the well-known Hermitian beam material stiffness. For mass-stiffness combinations, three objectives are noted: high accuracy for vibration analysis, wide separation of acoustic and optical branches, and low sensitivity to mesh distortion and boundary conditions. Only the first objective is examined in detail.

  Abstract
  Teaches by example the application of finite element templates in constructing high performance elements. The example discusses the improvement of the mass and geometric stiffness [...]

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• Partitioned formulation of internal fluid–structure interaction problems by localized Lagrange multipliers

  K. Park, C. Felippa, R. Ohayon

  Computer Methods in Applied Mechanics and Engineering (2001). Vol. 190 (24-25), pp. 2989-3007

  
  

  Abstract

  A partitioned formulation of compressible internal fluid–structure interaction problems is presented by employing a displacement model for both the fluid and structure. Partitioning is effected by a localized version of the method of Lagrange multipliers, which assigns two independent sets of Lagrange multipliers to the structural and fluid interfaces. Two major features of the present formulation include: an interface compliance normalization that helps capture the predominant physics of interaction phenomena when the interfaces are characterized by two radically different rigidities, and a novel transformation of the displacement model into a fluid-pressure model that is suitable for both transient and vibration analyses. The present formulation first solves for the interface Lagrange multipliers, which are subsequently used to solve for the structural displacements and the fluid displacement or pressure by employing two independent analysis modules.

  Abstract
  A partitioned formulation of compressible internal fluid–structure interaction problems is presented by employing a displacement model for both the fluid and structure. [...]

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• Partitioned analysis of coupled system

  C. Felippa, K. Park, C. Farhat

  Computer Methods in Applied Mechanics and Engineering (2001). Vol. 190 (s24-25), pp. 3247-3270

  
  

  Abstract

  This is a tutorial article that reviews the use of partitioned analysis procedures for the analysis of coupled dynamical systems. Attention is focused on the computational simulation of systems in which a structure is a major component. Important applications in that class are provided by thermomechanics, fluid–structure interaction and control–structure interaction. In the partitioned solution approach, systems are spatially decomposed into partitions. This decomposition is driven by physical or computational considerations. The solution is separately advanced in time over each partition. Interaction effects are accounted for by transmission and synchronization of coupled state variables. Recent developments in the use of this approach for multilevel decomposition aimed at massively parallel computation are discussed.

  Abstract
  This is a tutorial article that reviews the use of partitioned analysis procedures for the analysis of coupled dynamical systems. Attention is focused on the computational [...]

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• A historical outline of matrix structural analysis: a play in three acts

  C. Felippa

  Computers and Structures (2001). Vol. 79 (14), pp. 1313-1324

  
  

  Abstract

  The evolution of matrix structural analysis (MSA) from 1930 through 1970 is outlined. Highlighted are major contributions by Collar and Duncan, Argyris, and Turner, which shaped this evolution. To enliven the narrative the outline is configured as a three-act play. Act I describes the pre-WWII formative period. Act II spans a period of confusion during which matrix methods assumed bewildering complexity in response to conflicting demands and restrictions. Act III outlines the cleanup and consolidation driven by the appearance of the direct stiffness method, through which MSA completed morphing into the present implementation of the finite element method (FEM). No attempt is made at chronicling the more complex history of FEM itself.

  Abstract
  The evolution of matrix structural analysis (MSA) from 1930 through 1970 is outlined. Highlighted are major contributions by Collar and Duncan, Argyris, and Turner, which [...]

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• Fitting Strains and Displacements by Minimizing Dislocation Energy

  C. Felippa

  Computational and Technology Resources (2002). Paper 20

  
  

  Abstract

  We present a procedure for matching a displacement field to a given strain field, or vice- versa, over an arbitrary domain, which can be a finite element. The fitting criterion used is minimization of a dislocation energy functional. The strain field, whether given or fitted, need not be compatible. The method has four immediate applications: (i) finite element stiffness formulation based on fitting assumed-natural-strain (ANS) fields to node displacements; (ii) pointwise recovery of an internal displacement field in ANS elements as required for consistent mass, body load or geometric stiffness computations; (iii) recovery of smoothed strains from node displacements for stress post-processing, and (iv) system identification and damage detection from experimental data. The article focuses on application (i) for the strain fitting (SF) problem and (ii) for the displacement fitting (DF) problem. The separation of mean and deviatoric strains is emphasized whenever it is found convenient to simplify calculations.

  Abstract
  We present a procedure for matching a displacement field to a given strain field, or vice- versa, over an arbitrary domain, which can be a finite element. The fitting criterion [...]

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• The construction of free–free flexibility matrices for multilevel structural analysis

  C. Felippa, K. Park

  Computer Methods in Applied Mechanics and Engineering (2002). Vol. 191 (19-20), pp. 2139-2168

  
  

  Abstract

  This article considers a generalization of the classical structural flexibility matrix. It expands on previous papers by taking a deeper look at computational considerations at the substructure level. Direct or indirect computation of flexibilities as “influence coefficients” has traditionally required pre-removal of rigid body modes by imposing appropriate support conditions, mimicking experimental arrangements. With the method presented here the flexibility of an individual element or substructure is directly obtained as a particular generalized inverse of the free–free stiffness matrix. This generalized inverse preserves the stiffness spectrum. The definition is element independent and only involves access to the stiffness generated by a standard finite element program and the separate construction of an orthonormal rigid-body mode basis. The free–free flexibility has proven useful in special application areas of finite element structural analysis, notably massively parallel processing, model reduction and damage localization. It can be computed by solving sets of linear equations and does not require processing an eigenproblem or performing a singular value decomposition. If substructures contain thousands of d.o.f., exploitation of the stiffness sparseness is important. For that case this paper presents a computation procedure based on an exact penalty method, and a projected rank-regularized inverse stiffness with diagonal entries inserted by the sparse factorization process. These entries can be physically interpreted as penalty springs. This procedure takes advantage of the stiffness sparseness while forming the full free–free flexibility, or a boundary subset, and is backed by an in-depth null space analysis for robustness.

  Abstract
  This article considers a generalization of the classical structural flexibility matrix. It expands on previous papers by taking a deeper look at computational considerations [...]

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• A simple algorithm for localized construction of non‐matching structural interfaces

  K. Park, C. Felippa, G. Rebel

  International Journal for Numerical Methods in Engineering (2002). Vol. 53 (9), pp. 2117-2142

  
  

  Abstract

  A simple and effective algorithm for the modular construction of non-matched interfaces is presented for the partitioned solution of large-scale structural problems. The formulation is based on a recently developed four-field variational principle, which introduces a connection frame between the interfaced partitions. A key result of the present study is a frame nodal placement criterion that uniquely determines the frame discretization into piecewise linear elements so that the interface patch test condition is satisfied a priori. The method is demonstrated with several 2D and 3D example problems

  Abstract
  A simple and effective algorithm for the modular construction of non-matched interfaces is presented for the partitioned solution of large-scale structural problems. The formulation [...]

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