Theory Used in ADINA

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    For the theory used in ADINA, for structural analysis, CFD, and FSI, and also for the philosophy used in the program development, please refer to the publications given here:

  • Books by K.J. Bathe and co-authors



    To Enrich Life
    (Sample pages here)

  • Solutions to exercises in the book "Finite Element Procedures", 2nd Edition, 2014 are given in this manual (.pdf)

    The Chinese translation of the 2nd edition is also available: Vol. 1 Vol. 2

  • Proceedings edited by K.J. Bathe
  • CompFSM
    Computational Fluid and Solid Mechanics 2001-2011
    (6 volumes)


  • Theory and Modeling Guides distributed on the ADINA Installation CD.
    These manuals describe in short form the theory used in ADINA Structures, Thermal, CFD and EM, and give hints for modeling problems correctly. For ADINA users: manuals

  • Papers on the Development of Finite Element Methods, with some of these Papers Related to ADINA

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  • For publications that reference the use of ADINA, please see here.

    A Numerical Study of a Natural Convection Flow in a Cavity

    Guo, Yanhu; Bathe, Klaus-Jürgen. Source: Int. J. Num. Meth. in Fluids, v 40, 1045-1057, 2002.

    ISSN: 0271-2091 (print); 1097-0363 (online)

    Publisher: John Wiley & Sons, Ltd.

    Abstract: The solution to a benchmark problem of a differentially heated cavity flow with Ra=3.4×105 and Pr=0.71 was solicited for presentation at a special session entitled ‘Computational Predictability of Natural Convection Flows in Enclosures’ which was held at the First MIT Conference on Computational Fluid and Solid Mechanics. The objective of this paper is to present the results obtained using the ADINA System.The 9-node quadrilateral element in ADINA was used with various meshes. Periodic solutions with a period of 3.42-3.43 were obtained. Compared with the average values of the solution variables and the periods, the calculated amplitudes of the periodic solutions were found to be more sensitive to the spatial and temporal discretizations used. The flow patterns, such as boundary layers, vortices, etc. were also studied using a fine 40×120 element mesh. Vortices and their evolutions were revealed inside the periodic flow field.

    Keywords: natural convection, unsteady cavity flow, ADINA

    An Evaluation of the Lyapunov Characteristic Exponent of Chaotic Continuous Systems

    Rugonyi, Sandra; Bathe, Klaus-Jürgen. Source: Int. J. for Numerical Methods in Eng., v 56, 145-163, 2003.

    ISSN: 0029-5981 CODEN: IJNMBH

    Publisher: John Wiley & Sons, Ltd.

    Abstract: A procedure to calculate the Lyapunov characteristic exponent of the response of structural continuous systems, discretized using finite element methods, is proposed. The Lyapunov characteristic exponent can be used to characterize the asymptotic stability of the system dynamic response, and it is frequently employed to identify a chaotic behaviour. The proposed procedure can also be used in the stability characterization of fluid-structure interaction systems in which the focus of the analysis is on the behaviour of the structural part.

    Keywords: stability analysis, Lyapunov characteristic exponent, chaotic behaviour, non-linear dynamics, dynamic stability

    A Flow-Condition-Based Interpolation Mixed Finite Element Procedure for Higher Reynolds Number Fluid Flows

    Bathe, Klaus-Jürgen; Pontaza, Juan P. Source: Mathematical Models and Methods in Applied Sciences, v 12, no. 4, 525-539, 2002.

    ISSN: 0218-2025

    Publisher: World Scientific Publishing Company

    Abstract: We present a (somewhat) new finite element procedure for the analysis of higher Reynolds number fluid flows. While two-dimensional conditions and incompressible fluid flows are considered, the scheme can directly be used for three-dimensional conditions and also has good potential for compressible flow analysis. The procedure is based on the use of a nine-node element, optimal for incompressible analysis (the 9/3 or 9/4-c elements), and a Petrov–Galerkin formulation with exponential weight functions (test functions). These functions are established from the flow conditions along the edge- and mid-lines of the element. An important feature is that for low Reynolds number flow, the weight functions are the usual biquadratics and as higher Reynolds number flow is considered, the functions "automatically" skew so as to provide the necessary stability for the solution (the upwinding effect). Since the test functions are calculated by the flow conditions, no artificial constant is set by the analyst. The procedure is simple to implement. We present some solution experiences and conclude that while the procedure is not the "ideal" solution scheme sought, it has some valuable attributes and good potential for further improvements.

    Keywords: finite elements, incompressible fluid flows

    The Method of Finite Spheres with Improved Numerical Integration

    De, Suvranu; Bathe, Klaus-Jürgen. Source: Computers & Structures, v 79, 2183-2196, 2001.

    ISSN: 0045-7949 CODEN: CMSTCJ

    Publisher: Elsevier Ltd

    Abstract: The method of finite spheres was introduced as a truly meshless technique with the goal of achieving computational efficiency in a mesh-free procedure. In this paper we report several new numerical integration rules that result in a significant reduction in computational cost.

    Keywords: method of finite spheres, meshless, mesh-free procedure, numerical integration rules

    Stability of Finite Element Mixed Interpolations for Contact Problems

    Bathe, Klaus-Jürgen; Brezzi, Franco. Source: Proceedings della Accademia Nazionale dei Lincei, s. 9, v 12, 159-166, 2001.

    ISSN: N.A.

    Publisher: Accademia Nazionale dei Lincei

    Abstract: We consider the formulation of contact problems using a Lagrange multiplier to enforce the contact no-penetration constraint. The finite element discretization of the formulation must satisfy stability conditions which include an inf-sup condition. To identify which finite element interpolations in the contact constraint lead to stable ( and optimal ) numerical solutions we focus on the finite element discretization and solution of a ' simple' model problem. While a simple problem to avoid the need for technicalities, the analysis of the finite element discretizations to solve the problem gives valuable insight and allows quite general conclusions on the use of different interpolation schemes.

    Keywords: contact problems, elasticity, inf-sup condition

    Optimal Consistency Errors for General Shell Elements

    Chapelle, Dominique; Bathe, Klaus-Jürgen. Source: C. R. Acad. Sci. Paris, t.332, Serie I, 771-776, 2001.

    ISSN: 0764-4442

    Publisher: Académie des sciences/Éditions scientifiques et médicales Elsevier SAS.

    Abstract: We obtain estimates for the consistency errors arising in general shell element procedures, which are widely used in engineering practice. These estimates improve a previous result by the same authors. Moreover, numerical experiments indicate that these new estimates are optimal. Further, we introduce a modified procedure for which nominal convergence is recovered. These results are of much practical significance.

    Keywords: shell element, error estimate

    Towards an Efficient Meshless Computational Technique: the Method of Finite Spheres

    De, Suvranu; Bathe, Klaus-Jürgen. Source: Engineering Computations, v 18, 170-192, 2001

    ISSN: 0264-4401

    Publisher: MCB UP Ltd

    Abstract: Computational efficiency and reliability are clearly the most important requirements for the success of a meshless numerical technique. While the basic ideas of meshless techniques are simple and well understood, an effective meshless method is very difficult to develop. The efficiency depends on the proper choice of the interpolation scheme, numerical integration procedures and techniques of imposing the boundary conditions. These issues in the context of the method of finite spheres are discussed.

    Keywords: method of finite spheres, meshless, interpolation scheme

    Higher-Order-Accuracy Points in Isoparametric Finite Element Analysis and Application to Error Assessment

    Hiller, Jean-François; Bathe, Klaus-Jürgen. Source: Computers & Structures, v 79, 1275-1285, 2001.

    ISSN: 0045-7949 CODEN: CMSTCJ

    Publisher: Elsevier Ltd

    Abstract: We consider the systematic calculation of higher-order-accuracy points in finite element analysis. The cases of varying material properties and distortion of elements are included. The results obtained are used to establish an error estimator that is element-based and applicable to varying material properties. Some example problems are solved and show the applicability of the error estimator. While one-dimensional problems are studied in detail, the extension to two- and three-dimensional analysis is indicated.

    Keywords: finite element analysis, error estimator, material properties

    On the Finite Element Analysis of Fluid Flows Fully Coupled with Structural Interactions

    Rugonyi, Sandra; Bathe, Klaus-Jürgen. Source: Computer Modeling in Engineering & Sciences, v 2, 195-212, 2001.

    ISSN: 1526-1492 (printed); 1526-1506 (online)

    Publisher: Tech Science Press

    Abstract: The solution of fluid flows, modeled using the Navier-Stokes or Euler equations, fully coupled with structures/solids is considered. Simultaneous and partitioned solution procedures, used in the solution of the coupled equations, are briefly discussed, and advantages and disadvantages of their use are mentioned. In addition, a simplified stability analysis of the interface equations is presented, and unconditional stability for certain choices of time integration schemes is shown. Furthermore, the long-term dynamic stability of fluid-structure interaction systems is assessed by the use of Lyapunov characteristic exponents, which allow differentiating between a chaotic and a regular system behavior. Some state-of-the-art numerical solutions are also presented to indicate the type of problems that can now be solved using currently available techniques.

    Keywords: fluid-structure interaction, arbitrary Lagrangian-Eulerian formulation, finite element methods, coupled procedures, Lyapunov characteristic exponent, dynamic stability.

    Displacement/Pressure Mixed Interpolation in the Method of Finite Spheres

    De, Suvranu; Bathe, Klaus-Jürgen. Source: Int. J. for Numerical Methods in Eng., v 51, 275-292, 2001.

    ISSN: 0029-5981 CODEN: IJNMBH

    Publisher: John Wiley & Sons, Ltd

    Abstract: The displacement-based formulation of the method of finite spheres is observed to exhibit volumetric ‘locking’ when incompressible or nearly incompressible deformations are encountered. In this paper, we present a displacement/pressure mixed formulation as a solution to this problem. We analyse the stability and optimality of the formulation for several discretization schemes using numerical inf-sup tests. Issues concerning computational efficiency are also discussed.

    Keywords: method of finite spheres, meshless technique, incompressible analysis

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