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• A formulation based on localized Lagrange multipliers for BEM–FEM coupling in contact problems

  J. González, K. Park, C. Felippa, R. Abascal

  Computer Methods in Applied Mechanics and Engineering (2008). Vol. 197 (6-8), pp. 623-640

  
  

  Abstract

  This paper presents a unified formulation for the combination of the finite element method (FEM) and the boundary element method (BEM) in 3D frictional contact problems that is based on the use of localized Lagrange multipliers (LLMs). Resolution methods for the contact problem between non-matching meshes have traditionally been based on a direct coupling of the contacting solids using classical Lagrange multipliers. These methods tend to generate strongly coupled systems that require a deep knowledge of the discretization characteristics on each side of the contact zone complicating the process of mixing different numerical techniques. In this work a displacement contact frame is inserted between the FE and BE interface meshes, discretized and finally connected to the contacting substructures using LLMs collocated at the msh-interface nodes. This methodology will provide a partitioned formulation which preserves software modularity and facilitates the connection of non-matching FE and BE meshes.  

  Abstract
  This paper presents a unified formulation for the combination of the finite element method (FEM) and the boundary element method (BEM) in 3D frictional contact problems [...]

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• Treatment of acoustic fluid–structure interaction by localized Lagrange multipliers: Formulation

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

  Computer Methods in Applied Mechanics and Engineering (2008). Vol. 197 (33-40), pp. 3057-3079

  
  

  Abstract

  A new concept is presented for modeling the dynamic interaction between an acoustic fluid and an elastic structure. The coupling of this multiphysics system is done by inserting a kinematic interface frame between the fluid and the structure, and using node-collocated Lagrange multipliers to connect the frame to each subsystem. The time-domain response analysis is performed by a partitioned analysis procedure. The main advantages of this localized Lagrange multiplier (LLM) primal-dual coupling method are: complete localization of the structure and fluid subsystems, elimination of the conventional predictor in the partitioned time integration method, and the ability to accommodate non-matching meshes. The standard Newmark time integrator is used on both the fluid and structure models. It is shown that if the integrator is A-stable and second-order accurate for a monolithic treatment, it retains those properties for both Mortar and LLM partitioned solution procedures. Infinite and finite piston problems are used to explain and verify the methodology. A sequel paper under preparation presents and discusses a set of benchmark and application examples that involve the response of existing dams to seismic excitation.

  Abstract
  A new concept is presented for modeling the dynamic interaction between an acoustic fluid and an elastic structure. The coupling of this multiphysics system is done by inserting [...]

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• Plastic buckling and collapse of thin shell structures, using layered plastic modeling and co-rotational ANDES finite elements

  N. Cortivo, C. Felippa, H. Bavestrello, W. Silva

  Computer Methods in Applied Mechanics and Engineering (2009). Vol. 198 (5-8), pp. 758-798

  
  

  Abstract

  This study reveals an analysis of plastic buckling and collapse of thin shell structures. For this purpose, the co-rotational and layered plastic model as well as ANDES (Assumed Natural Deviatoric Strain) finite element formulations are used. The co-rotational kinematics formulation splits the translational and rotational deformations in a small deformation analysis. The ANDES finite element is modified to elastoplastic ANDES finite element by the introduction of the von Mises yield criterion elastoplastic formulation on its original deformation model. In order to accommodate the plasticity formulation, the Gauss point layered integration is inserted through of thickness of the element to produce the internal force vector and material stiffness matrix. Special effort is devoted to maintain the consistency of the internal forces and tangent stiffness as well as to enhance the robustness of element level computations. The arc-length method is used to follow the postbuckling equilibrium path. Results are presented for several benchmark elastoplastic shell problems available in the present literature, which are generally in agreement with the present work

  Abstract
  This study reveals an analysis of plastic buckling and collapse of thin shell structures. For this purpose, the co-rotational and layered plastic model as well as ANDES (Assumed [...]

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• The d'Alembert–Lagrange principal equations and applications to floating flexible systems

  K. Park, C. Felippa, R. Ohayon

  International Journal for Numerical Methods in Engineering (2009). Vol. 77 (8), pp. 1072-1099

  
  

  Abstract

  This paper addresses the dynamics and quasi‐statics of floating flexible structures as well as extensions to unconstrained substructures and partitions of coupled mechanical systems. The principal solution is defined as the state of self‐equilibrated forces obtained as the particular solution of the rigid motion and interface equilibrium equations. This solution is independent of the stress–strain constitutive properties as well as of the compatibility equations. For statically determinate systems, the principal solution is the final force solution. For statically indeterminate systems, the correction due to flexibility and compatibility is orthogonal to the principal solution. The formulation is done in the context of d'Alembert's principle, which supplies the d 'Alembert–Lagrange principal equations for floating bodies. These are obtained by summation of virtually working forces and moments acting on the floating systems. Applications of this approach are demonstrated on a set of dynamic and quasi‐static example problems of increasing generality. Linkage to variational principles with an interface potential is eventually discussed as providing the theoretical foundation for handling interacting semi‐discrete subsystems linked by node‐collocated Lagrange multipliers.

  Abstract
  This paper addresses the dynamics and quasi‐statics of floating flexible structures as well as extensions to unconstrained substructures and partitions of coupled mechanical [...]

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• Partitioned formulation of internal and gravity waves interacting with flexible structures

  K. Park, R. Ohayon, C. Felippa, J. González

  Computer Methods in Applied Mechanics and Engineering (2010). Vol. 199 (9-12), pp. 723-733

  
  

  Abstract

  This paper presents a partitioned modeling of internal and gravity fluid waves that interact with flexible structures. The governing interaction model consists of three completely partitioned entities: fluid model, structural model, and interface model that acts as an internal constraint on the fluid–structure interface boundary. Thus, the proposed partitioned multi-physics modeling can employ two completely modular fluid and structure software modules plus an interface solver, hence amenable to partitioned solution algorithms. The interface discretization can exploit the nonmatching interface algorithm previously developed via the method of localized Lagrange multipliers. Also noted is that the present fluid model can make use of widely available finite element software for standard Poisson-type problems.

  Abstract
  This paper presents a partitioned modeling of internal and gravity fluid waves that interact with flexible structures. The governing interaction model consists of three [...]

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• Partitioned vibration analysis of internal fluid-structure interaction problems

  J. González, K. Park, I. Lee, C. Felippa, R. Ohayon

  International Journal for Numerical Methods in Engineering (2012). Vol. 92 (3), pp. 260-300

  
  

  Abstract

  A partitioned, continuum‐based, internal fluid–structure interaction (FSI) formulation is developed for modeling combined sloshing, acoustic waves, and the presence of an initial pressurized state. The present formulation and its computer implementation use the method of localized Lagrange multipliers to treat both matching and non‐matching interfaces. It is shown that, with the context of continuum Lagrangian kinematics, the fluid sloshing and acoustic stiffness terms originate from an initial pressure term akin to that responsible for geometric stiffness effects in solid mechanics. The present formulation is applicable to both linearized vibration analysis and nonlinear FSI transient analysis provided that a convected kinematics is adopted for updating the mesh geometry in a finite element discretization. Numerical examples illustrate the capability of the present procedure for solving coupled vibration and nonlinear sloshing problems.

  Abstract
  A partitioned, continuum‐based, internal fluid–structure interaction (FSI) formulation is developed for modeling combined sloshing, acoustic waves, and the presence [...]

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• Reusing linear finite elements in material and geometrically nonlinear analysis—Application to plane stress problems

  M. Gholizadeh, M. Sivaselvan, C. Felippa

  Finite Elements in Analysis and Desing (2013). Vol. 69, pp. 62-72

  
  

  Abstract

  This paper describes a computational approach suitable for combined material and geometrically nonlinear analysis by the Finite Element Method. Its main advantage is reuse: once a finite element has been developed with good performance in linear analysis, extension to material and geometrically nonlinear problems is simplified. Extension to geometrically nonlinear problems is enabled by a corotational kinematic description, and that to material nonlinear problems by an optimization-based solution algorithm. The approach thus comprises three ingredients—the development of a high performance linear finite element (for example, using the ANDES concept), a corotational kinematic description, and an optimization algorithm. The main constraint in the application of the corotational formulation is restriction to small deformational displacements. The paper illustrates the realization of the three ingredients on plane stress problems that exhibit elasto-plastic material behavior. Numerical examples are presented to illustrate the effectiveness of the approach. Comparison is made with respect to solutions provided by the commercial nonlinear code ABAQUS as reference.

  Abstract
  This paper describes a computational approach suitable for combined material and geometrically nonlinear analysis by the Finite Element Method. Its main advantage is reuse: [...]

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• A solid‐shell corotational element based on ANDES, ANS and EAS for geometrically nonlinear structural analysis

  M. Gholizadeh, M. Sivaselvan, C. Felippa

  International Journal for Numerical Methods in Engineering (2013). Vol. 95 (2), pp. 145-180

  
  

  Abstract

  This paper describes an eight‐node, assumed strain, solid‐shell, corotational element for geometrically nonlinear structural analysis. The locally linear kinematics of the element is separated into in‐plane (which is further decoupled into membrane and bending), thickness and transverse shear components. This separation allows using any type of membrane quadrilateral formulation for the in‐plane response. Assumed strain fields for the three components are constructed using different approaches. The Assumed Natural Deviatoric Strain approach is used for the in‐plane response, whereas the Assumed Natural Strain approach is used for the thickness and transverse shear components. A strain enhancement based on Enhanced Assumed Strain concepts is also used for the thickness component. The resulting element passes well‐known shell element patch tests and exhibits good performance in a number of challenging benchmark tests. The formulation is extended to the geometric nonlinear regime using an element‐independent corotational approach. Some key properties of the corotational kinematic description are discussed. The element is tested in several well‐known shell benchmarks and compared with other thin‐shell and solid‐shell elements available in the literature, as well as with commercial nonlinear FEM codes.

  Abstract
  This paper describes an eight‐node, assumed strain, solid‐shell, corotational element for geometrically nonlinear structural analysis. The locally linear kinematics of [...]

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• Mass Matrix Templates: General Description and 1D Examples

  C. Felippa, Q. Guo, K. Park

  Archives of Computational Methods in Engineering (2015). Vol. 22 (1), pp. 1-65

  
  

  Abstract

  This article is a tutorial exposition of the template approach to the construction of customized mass-stiffness pairs for selected applications in structural dynamics. The main focus is on adjusting the mass matrix. Two well known discretization methods, described in FEM textbooks since the late 1960s, lead to diagonally lumped and consistent mass matrices, respectively. Those models are sufficient to cover many engineering applications but for some problems 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. 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. 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. The paper provides a general overview of the topic, and illustrates it with one-dimensional structural elements: bars and beams.

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

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• Inverse mass matrix via the method of localized lagrange multipliers

  J. González, R. Kolman, S. Cho, C. Felippa, K. Park

  International Journal for Numerical Methods in Engineering (2017). Vol. 113 (2), pp. 277-295

  
  

  Abstract

  An efficient method for generating the mass matrix inverse of structural dynamic problems is presented, which can be tailored to improve the accuracy of target frequency ranges and/or wave contents. The present method bypasses the use of biorthogonal construction of a kernel inverse mass matrix that requires special procedures for boundary conditions and free edges or surfaces and constructs the free‐free inverse mass matrix using the standard FEM procedure. The various boundary conditions are realized by the the method of localized Lagrange multipliers. In particular, the present paper constructs the kernel inverse matrix by using the standard FEM elemental mass matrices. It is shown that the accuracy of the present inverse mass matrix is almost identical to that of a conventional consistent mass matrix or a combination of lumped and consistent mass matrices. Numerical experiments with the proposed inverse mass matrix are conducted to validate its effectiveness when applied to vibration analysis of bars, beams, and plain stress problems.  

  Abstract
  An efficient method for generating the mass matrix inverse of structural dynamic problems is presented, which can be tailored to improve the accuracy of target frequency [...]

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