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.

    3D-shell elements for structures in large strains

    Sussman, Theodore, and Bathe, Klaus-Jürgen. Source: Computers & Structures, In Press

    Abstract: We present in this paper MITC shell elements for large strain solutions of shell structures. While we focus on the 4-node element, the same formulation is also applicable to the 3-node element. Since the elements are formulated using three-dimensional continuum theory with the full three-dimensional constitutive behavior, they are referred to as 3D-shell elements. Specific contributions in this paper are that the elements are formulated using two control vectors at each node to describe the large deformations, MITC tying and volume preserving conditions acting directly on the material fiber vectors to avoid shear locking, and a pressure interpolation to circumvent volumetric locking. Also, we present solutions to some large strain shell problems that represent valuable benchmark tests for any large strain shell analysis capability.

    ISSN: 0045-7949 CODEN: CMSTCJ

    Publisher: Elsevier Ltd

    Keywords: Shell elements, 3D-shell elements, MITC tying, large strains, benchmark solutions, buckling

    The finite element method enriched by interpolation covers

    Kim, Jaehyung, and Bathe, Klaus-Jürgen. Source: Computers & Structures, v. 116, 35-49, 2013

    Abstract: In this paper, we focus on an enriched finite element solution procedure for low-order elements based on the use of interpolation cover functions. We consider the 3-node triangular and 4-node tetrahedral displacement-based elements for two- and three-dimensional analyses, respectively. The standard finite element shape functions are used with interpolation cover functions over patches of elements to increase the convergence of the finite element scheme. The cover functions not only capture higher gradients of a field variable but also smooth out inter-element stress jumps. Since the order of the interpolations in the covers can vary, the method provides flexibility to use different covers for different patches and increases the solution accuracy without any local mesh refinement. As pointed out, the procedure can be derived from various general theoretical approaches and the basic theory has been presented earlier. We evaluate the effectiveness of the method, and illustrate the power of the scheme through the solution of various problems. The method also has potential for the development of error measures.

    ISSN: 0045-7949 CODEN: CMSTCJ

    Publisher: Elsevier Ltd

    Keywords: Enriched finite elements, 3-node 2D & 4-node 3D elements, cover functions, higher convergence, increase in accuracy, adaptive interpolation

    A stress improvement procedure

    Payen, Daniel Jose, and Bathe, Klaus-Jürgen. Source: Computers & Structures, v. 112-113, 311-326, 2012

    Abstract: In this paper, we present a novel procedure to improve the stress predictions in static, dynamic and non-linear analyses of solids. We focus on the use of low-order displacement-based finite elements – 3-node and 4-node elements in two-dimensional (2D) solutions, and 4-node and 8-node elements in 3D solutions – because these elements are computationally efficient provided good stress convergence is obtained. We give a variational basis of the new procedure and compare the scheme, and its performance, with other effective previously proposed stress improvement techniques. We observe that the stresses of the new procedure converge quadratically in 1D and 2D solutions, i.e. with the same order as the displacements, and conclude that the new procedure shows much promise for the analysis of solids, structures and multiphysics problems, to calculate improved stress predictions and to establish error measures.

    ISSN: 0045-7949 CODEN: CMSTCJ

    Publisher: Elsevier Ltd

    Keywords: Finite element stress calculation, low-order displacement-based elements, stress recovery, stress convergence, mixed method, error measures

    The subspace iteration method – Revisited

    Bathe, Klaus-Jürgen. Source: Computers & Structures, 10.1016/j.compstruct.2012.06.002

    Abstract: The objective in this paper is to present some recent developments regarding the subspace iteration method for the solution of frequencies and mode shapes. The developments pertain to speeding up the basic subspace iteration method by choosing an effective number of iteration vectors and by the use of parallel processing. The subspace iteration method lends itself particularly well to shared and distributed memory processing. We present the algorithms used and illustrative sample solutions. The present paper may be regarded as an addendum to the publications presented in the early 1970s, see Refs.[1,2], taking into account the changes in computers that have taken place.

    ISSN: 0045-7949 CODEN: CMSTCJ

    Publisher: Elsevier Ltd

    Keywords: Frequencies, mode shapes, subspace iteration method, parallel processing, shared memory, distributed memory

    Insight into an implicit time integration scheme for structural dynamics

    Bathe, Klaus-Jürgen and Noh, Gunwoo. Source: Computers & Structures, v. 98-99, 1-6, 2012.

    Abstract: In Refs. [1,2], an effective implicit time integration scheme was proposed for the finite element solution of nonlinear problems in structural dynamics. Various important attributes were demonstrated. In particular, it was shown that the scheme remains stable, without the use of adjustable parameters, when the commonly used trapezoidal rule results in unstable solutions. In this paper we focus on additional important attributes of the scheme, and specifically on showing that the procedure can also be effective in linear analyses. We give, in comparison to other methods, the spectral radius, period elongation, and amplitude decay of the scheme and study the solution of a simple ‘model problem’ with a very flexible and stiff response.

    ISSN: 0045-7949 CODEN: CMSTCJ

    Publisher: Elsevier Ltd

    Keywords: Structural dynamics, finite elements, implicit time integration, trapezoidal rule, Newmark method, Bathe method

    Crushing and crashing of tubes with implicit time integration

    Kazanci, Zafer, and Bathe, Klaus-Jürgen. Source: International Journal of Impact Engineering, v. 42, 80-88, 2012.

    Abstract: The axial crushing and crashing of thin-walled high-strength steel tubes is performed using 3D-shell finite elements and an implicit time integration scheme. The calculated results are compared with published experimental data and results obtained using explicit time integration. The objective is to show that, while for such analyses generally explicit time integration is used, with the current state of the art also an implicit time integration solution should be considered, and such solution approach can provide an effective alternative for a simulation.

    ISSN: 0734-743X

    Publisher: Elsevier Ltd

    Keywords: Dynamic axial crushing, implicit time integration, Bathe method, 3D-shell element, crash tube

    A finite element method enriched for wave propagation problems

    Ham, Seounghyun, and Bathe, Klaus-Jürgen. Source: Computers & Structures, v. 94-95, 1-12, 2012.

    Abstract: An enriched finite element method is presented to solve various wave propagation problems. The proposed method is an extension of the procedure introduced by Kohno, Bathe, and Wright for one-dimensional problems [1]. Specifically, the novelties are: two-dimensional problems are solved (and three-dimensional problems would be tackled similarly), a scheme is given to overcome ill-conditioning, the method is presented for time-dependent problems, and focus is on the solution of problems in solids and structures using real arithmetic only. The method combines advantages of finite element and spectral techniques, but an important point is that it preserves the fundamental properties of the finite element method. The general formulation of the procedure is given and various examples are solved to illustrate the capabilities of the proposed scheme.

    ISSN: 0045-7949 CODEN: CMSTCJ

    Publisher: Elsevier Ltd

    Keywords: Wave propagation, finite elements, spectral methods, harmonics

    Improved stresses for the 4-node tetrahedral element

    Payen, Daniel Jose and Bathe, Klaus-Jürgen. Source: Computers & Structures, v. 89, 1265-1273, 2011.

    Abstract: The objective in this paper is to present the method for the calculation of improved stresses published by Payen and Bathe in [1] for the 4-node three-dimensional tetrahedral element. This element is widely used in engineering practice to obtain, in general, only ‘‘guiding’’ results in the analysis of solids because the element is known to be poor in stress predictions. We show in this paper the potential of this novel approach to significantly enhance the stress predictions with the 4-node tetrahedral element at a relatively low computational cost.

    ISSN: 0045-7949 CODEN: CMSTCJ

    Publisher: Elsevier Ltd

    Keywords: Stress improvements, stress recovery, NPF-based stresses, nodal point forces, finite elements 4-Node tetrahedral element

    Modeling large strain anisotropic elasto-plasticity with logarithmic strain and stress measures

    Caminero, Miguel Ángel, Montáns, Francisco Javier and Bathe, Klaus-Jürgen. Source: Computers & Structures, v. 89, 826-843, 2011.

    Abstract: In this paper we present a model and a fully implicit algorithm for large strain anisotropic elasto- plasticity with mixed hardening in which the elastic anisotropy is taken into account. The formulation is developed using hyperelasticity in terms of logarithmic strains, the multiplicative decomposition of the deformation gradient into an elastic and a plastic part, and the exponential mapping. The novelty in the computational procedure is that it retains the conceptual simplicity of the large strain isotropic elasto- plastic algorithms based on the same ingredients. The plastic correction is performed using a standard small strain procedure in which the stresses are interpreted as generalized Kirchhoff stresses and the strains as logarithmic strains, and the large strain kinematics is reduced to a geometric pre- and post- processor. The procedure is independent of the specified yield function and type of hardening used, and for isotropic elasticity, the algorithm of Eterovic and Bathe is automatically recovered as a special case. The results of some illustrative finite element solutions are given in order to demonstrate the capabilities of the algorithm.

    ISSN: 0045-7949 CODEN: CMSTCJ

    Publisher: Elsevier Ltd

    Keywords: Computational elasto-plasticity, large strains, logarithmic strains, hyperelasticity, mixed hardening

    The MITC9 shell element in plate bending: mathematical analysis of a simplified case

    Bathe, Klaus-Jürgen, Brezzi, Franco and Marini, L. Donatella. Source: Comput. Mech., v. 47, 617–626, 2011.

    Abstract: We consider the 9-node shell element referred to as the MITC9 shell element in plate bending solutions and present a simplified mathematical analysis. The element uses bi-quadratic interpolations of the rotations and transverse displacement, and the “rotated Raviart-Thomas” interpolations for the transverse shear stresses. A rigorous mathematical analysis of the element is still lacking, even for the simplified case of plate solutions (that is, flat shells), although the numerical evidence suggests a good and reliable behavior. Here we start such an analysis by considering a very simple particular case; namely, a rectangular plate, clamped all around the boundary, and solved with a uniform decomposition. Moreover, we consider only the so-called limit case, corresponding to the limit equations that are obtained for the thickness t going to zero. While the mathematical analysis of the limit case is simpler, such analysis, in general, gives an excellent indication of whether shear locking is present in the real case t > 0. We detail that the element in the setting considered shows indeed optimal behavior.

    ISSN: 0178-7675

    Publisher: Springer-Verlag

    Keywords: Plate, shell, shear locking, MITC9 shell element, convergence analysis

    The use of nodal point forces to improve element stresses

    Payen, Daniel Jose and Bathe, Klaus-Jürgen. Source: Computers & Structures, v. 89, 485-495, 2011.

    Abstract: We present in this paper a novel approach to stress calculations in finite element analysis. Rather than using the stress assumption employed in establishing the stiffness matrix, the element nodal point forces are used, in a simple way, to enhance the finite element stress predictions at a low computational cost. While this paper focuses on the improvement of the stress accuracy, the proposed procedure can also be used as a basis for error estimation. Moreover, the procedure is quite general, and has the potential for many applications in finite element analysis.

    ISSN: 0045-7949 CODEN: CMSTCJ

    Publisher: Elsevier Ltd

    Keywords: Stress improvements, stress recovery, nodal point forces, error estimation, finite elements

    Measuring the convergence behavior of shell analysis schemes

    Bathe, Klaus-Jürgen and Lee, Phill-Seung. Source: Computers & Structures, v. 89, 285-301, 2011.

    Abstract: While shells have been analyzed abundantly for many years in engineering and the sciences, improved finite element and related analysis methods are still much desired and researched. More general and effective finite element procedures are needed for complex shell structures, including for the analysis of composite shells and the optimization of shells. In this paper we discuss how finite element methods, and other analysis techniques, should be tested in order to identify their reliability and effectiveness. We summarize some important theoretical results, present appropriate test problems and convergence measures, and we illustrate our discussion through some novel numerical results. An important conclusion is that the testing has to be performed very carefully in order to obtain relevant results, and we show how this is accomplished in detail.

    ISSN: 0045-7949 CODEN: CMSTCJ

    Publisher: Elsevier Ltd

    Keywords: Shells, finite element, convergence, benchmark problems

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