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• A contact formulation based on localized Lagrange multipliers: Formulation and application to two-dimensional problems

  G. Rebel, K. Park, C. Felippa

  International Journal for Numerical Methods in Engineering (2002). Vol. 54 (2), pp. 263-297

  
  

  Abstract

  The non-penetration condition in contact problems is traditionally based on the classical Lagrange multiplier method. This method makes extensive use of modelling details of the contacting bodies for contact enforcement as the contact surface meshes are in general non-matching. To deal with this problem we introduce a novel element in the Lagrange multiplier approach of contact modelling, namely, a contact frame placed in between contacting bodies. It acts as a medium through which contact forces are transferred without violating equilibrium in the contact domain for discrete contact models. Only nodal information of the contacting bodies is required which makes the proposed contact enforcement generic. The contact frame has its own independent freedoms, which allows the formulation to pass contact patch tests by design.

  Abstract
  The non-penetration condition in contact problems is traditionally based on the classical Lagrange multiplier method. This method makes extensive use of modelling details [...]

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• A study of optimal membrane triangles with drilling freedoms

  C. Felippa

  Computer Methods in Applied Mechanics and Engineering (2003). Vol. 192 (16-18), pp. 2125-2168

  
  

  Abstract

  This article compares derivation methods for constructing optimal membrane triangles with corner drilling freedoms. The term “optimal” is used in the sense of exact inplane pure-bending response of rectangular mesh units of arbitrary aspect ratio. Following a comparative summary of element formulation approaches, the construction of an optimal three-node triangle using the ANDES formulation is presented. The construction is based upon techniques developed by 1991 in student term projects, but taking advantage of the more general framework of templates developed since. The optimal element that fits the ANDES template is shown to be unique if energy orthogonality constraints are enforced. Two other formulations are examined and compared with the optimal model. Retrofitting the conventional linear strain triangle element by midpoint-migrating and congruential transformations is shown unable to produce an optimal element, while rank deficiency is inevitable. Use of the quadratic strain field of the 1988 Allman triangle, or linear filtered versions thereof, is also unable to reproduce the optimal element. Moreover these elements exhibit serious aspect ratio lock. These predictions are verified on benchmark examples.

  Abstract
  This article compares derivation methods for constructing optimal membrane triangles with corner drilling freedoms. The term “optimal” is used in the sense [...]

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• A distortion-insensitive four-noded membrane quadrilateral that passes the patch test

  C. Felippa

  (2003). Proceedings Second MIT Conference on Compurational Fluid and Solid Mechanics, Pages 254-257

  
  

  Abstract

  This chapter discusses the possibility to develop a quadrilateral called Unidirectional Bending-Optimal Quadrilateral Panel (UBOQP) that passes the patch test for any geometry while being bending exact along one direction. This model is directly constructed using the framework of finite element templates by energy fitting arguments. UBOQP is not obtainable by any conventional finite element method (FEM) formulation, which illustrates the customization power of templates. Numerical experiments in a well-known mesh distortion test verify the exact response to bending for any material law, aspect ratio, and skew angles. The four-noded membrane (plane stress) flat quadrilateral is one of the most interesting models from the standpoint of its impact on both theory and practice of the FEM. The rectangular specialization is one of the two multidimensional finite element stiffnesses that is explicitly derived by Turner. It is widely believed that a four-noded flat quadrilateral membrane element that passes the patch test must be sensitive to the mesh distortion response to inplane bending actions.

  Abstract
  This chapter discusses the possibility to develop a quadrilateral called Unidirectional Bending-Optimal Quadrilateral Panel (UBOQP) that passes the patch test for any geometry [...]

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• A compendium of FEM integration formulas for symbolic work

  C. Felippa

  Engineering Computations (2004). Vol. 21 (8), pp. 867-890

  
  

  Abstract

  This paper presents a set of Mathematica modules that organizes numerical integration rules considered useful for finite element work. Seven regions are considered: line segments, triangles, quadrilaterals, tetrahedral, wedges, pyramids and hexahedra. Information can be returned in floating-point (numerical) form, or in exact symbolic form. The latter is useful for computer-algebra aided FEM work that carries along symbolic variables. A few quadrature rules were extracted from sources in the FEM and computational mathematics literature, and placed in symbolic form using Mathematica to generate own code. A larger class of formulas, previously known only numerically, were directly obtained through symbolic computations. Some unpublished non-product rules for pyramid regions were found and included in the collection. For certain regions: quadrilaterals, wedges and hexahedra, only product rules were included to economize programming. The collection embodies most FEM-useful formulas of low and moderate order for the seven regions noted above. Some gaps as regard region geometries and omission of non-product rules are noted in the conclusions. The collection may be used “as is” in support of symbolic FEM work thus avoiding contamination with floating arithmetic that precludes simplification. It can also be used as generator for low-level floating-point code modules in Fortran or C. Floating point accuracy can be selected arbitrarily. No similar modular collection applicable to a range of FEM work, whether symbolic or numeric, has been published before

  Abstract
  This paper presents a set of Mathematica modules that organizes numerical integration rules considered useful for finite element work. Seven regions are considered: line segments, [...]

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• A unified formulation of small-strain corotational finite elements: I. Theory

  C. Felippa, B. Haugen

  Computer Methods in Applied Mechanics and Engineering (2005). Vol. 194 (21-24), pp. 2285-2335

  
  

  Abstract

  This paper presents a unified theoretical framework for the corotational (CR) formulation of finite elements in geometrically nonlinear structural analysis. The key assumptions behind CR are: (i) strains from a corotated configuration are small while (ii) the magnitude of rotations from a base configuration is not restricted. Following a historical outline the basic steps of the element independent CR formulation are presented. The element internal force and consistent tangent stiffness matrix are derived by taking variations of the internal energy with respect to nodal freedoms. It is shown that this framework permits the derivation of a set of CR variants through selective simplifications. This set includes some previously used by other investigators. The different variants are compared with respect to a set of desirable qualities, including self-equilibrium in the deformed configuration, tangent stiffness consistency, invariance, symmetrizability, and element independence. We discuss the main benefits of the CR formulation as well as its modeling limitations.

  Abstract
  This paper presents a unified theoretical framework for the corotational (CR) formulation of finite elements in geometrically nonlinear structural analysis. The key assumptions [...]

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• Partitioned formulation of frictional contact problems using localized Lagrange multipliers

  J. González, K. Park, C. Felippa

  Computer Methods in Applied Mechanics and Engineering (2005). Vol. 22 (4), pp. 319-333

  
  

  Abstract

  Interface treatment methods for the contact problem between non-matching meshes have traditionally been based on a direct coupling of the contacting solids employing a master–slave strategy or classical Lagrange multipliers. These methods tend to generate strongly coupled systems that is dependent on the discretization characteristics on each side of the contact zone. In this work a displacement frame is intercalated between the interface meshes. The frame is then discretized so that the discrete frame nodes are connected to the contacting substructures using the localized Lagrange multipliers collocated at the interface nodes. The resulting methodology alleviates the need for master–slave book-keeping and provides a partitioned formulation which preserves software modularity, facilitates non-matching mesh treatment and passes the contact patch test. Frictional contact problems are used to demonstrate the salient features of the proposed method.

  Abstract
  Interface treatment methods for the contact problem between non-matching meshes have traditionally been based on a direct coupling of the contacting solids employing a master–slave [...]

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• Supernatural QUAD4: A template formulation

  C. Felippa

  Computer Methods in Applied Mechanics and Engineering (2006). Vol. 195 (41-43), pp. 5316-5342

  
  

  Abstract

  A universal template for a 4-node quadrilateral in plane stress is constructed using a combination of old and new techniques. The use of natural quantities, strongly advocated by J.H. Argyris since his Continua and Discontinua 1965 exposition, is further expanded. The qualifier ‘supernatural’ means that all governing equations: kinematic, constitutive and equilibrium, are expressed in both Cartesian and natural forms. These two sets are used to build different components of the template. With timely help from a computer algebra system, a template that include all possible quadrilateral elements that pass the Individual Element Test of Bergan and Hanssen emerges. It yields an infinite number of hitherto undiscovered instances that may be customized to fit special needs. A striking example is the construction of a two-trapezoid macroelement that is bending exact about one direction, for any amount of distortion. This concludes a five decade search that begins with the formulation of the wing-cover rectangular panel in Argyris’ 1954 seminal serial on Energy Methods and Structural Analysis.

  Abstract
  A universal template for a 4-node quadrilateral in plane stress is constructed using a combination of old and new techniques. The use of natural quantities, strongly advocated [...]

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• Construction of Customized Mass-Stiffness Pairs Using Templates

  C. Felippa

  Journal of Aerospace Engineering (2006). Vol. 19 (4), pp. 241-258

  
  

  Abstract

  This paper is a tutorial exposition of the template approach to the construction of customized mass-stiffness pairs for selected applications in structural dynamics. The exposition focuses on adjusting the mass matrix while a separately provided stiffness matrix is kept fixed. Two well known kinetic-energy discretization methods described in finite-element method (FEM) textbooks since the mid-1960s lead to diagonally lumped and consistent mass matrices, respectively. These two models are sufficient to cover many engineering applications. Occasionally, however, they fall short. The gap can be filled with a more general approach that relies on the use of templates. These are algebraic forms that carry free parameters. This approach is discussed in this paper using one-dimensional structural elements as examples. Templates have the virtue of producing a set of mass matrices that satisfy certain a priori constraint conditions such as symmetry, nonnegativity, invariance, and momentum conservation. In particular, the diagonally lumped and consistent versions can be obtained as instances. Thus those standard models are not excluded. Availability of free parameters, however, allows the mass matrix to be customized to special needs, such as high precision vibration frequencies or minimally dispersive wave propagation. An attractive feature of templates for FEM programming is that only one element implementation as module with free parameters is needed, and need not be recoded when the application problem class changes.

  Abstract
  This paper is a tutorial exposition of the template approach to the construction of customized mass-stiffness pairs for selected applications in structural dynamics. The exposition [...]

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• FEM and BEM coupling in elastostatics using localized Lagrange multipliers

  J. González, K. Park, C. Felippa

  International Journal for Numerical Methods in Engineering (2007). Vol. 69 (10), pp. 2058-2074

  
  

  Abstract

  The equations produced by the finite and boundary element methods in structural mechanics are expressed in different variables and cannot be linked without modifications. Conventional coupling methods of these numerical techniques have traditionally been based on the use of classical Lagrange multipliers making a direct connection of the two solids through their common interfaces using matching meshes and altering the formulation of one of the methods to make it compatible with the other. In this work, a discrete surface called frame is interposed between the connected subdomains to approximate their common interface displacements, it is treated using a finite element discretization and connected to each substructure using localized Lagrange multipliers collocated at the interface nodes. This methodology facilitates the connection of non-matching finite and boundary element meshes avoiding modifications to the numerical methods used and providing a partitioned formulation, which preserves software modularity.

  Abstract
  The equations produced by the finite and boundary element methods in structural mechanics are expressed in different variables and cannot be linked without modifications. [...]

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• Treatment of acoustic fluid–structure interaction by localized Lagrange multipliers and comparison to alternative interface-coupling methods

  M. Ross, M. Sprague, C. Felippa, K. Park

  Computer Methods in Applied Mechanics and Engineering (2009). Vol. 198 (9-12), pp. 986-1005

  
  

  Abstract

  This paper is a sequel on the topic of localized Lagrange multipliers (LLM) for applications of fluid–structure interaction (FSI) between finite-element models of an acoustic fluid and an elastic structure. The prequel paper formulated the spatial-discretization methods, the LLM interface treatment, the time-marching partitioned analysis procedures, and the application to 1D benchmark problems. Here, we expand on formulation aspects required for successful application to more realistic 2D and 3D problems. Additional topics include duality relations at the fluid–structure interface, partitioned vibration analysis, reduced-order modeling, handling of curved interface surfaces, and comparison of LLM with other coupling methods. Emphasis is given to non-matching fluid–structure meshes. We present benchmark examples that illustrate the benefits and drawbacks of competing interface treatments. Realistic application problems involving the seismic response of two existing dams are considered. These include 2D modal analyses of the Koyna gravity dam, transient-response analyses of that dam with and without reduced-order modeling, incorporation of nonlinear cavitation effects, and the 3D transient-response analysis of the Morrow Point arch dam

  Abstract
  This paper is a sequel on the topic of localized Lagrange multipliers (LLM) for applications of fluid–structure interaction (FSI) between finite-element models of an [...]

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