General Finite Element System Research Articles

Transient response of stochastic finite element systems using Dynamic Variability Response Functions

In this study a methodology is presented for effective analysis of dynamic systems with stochastic material properties. The concept of dynamic mean and variability response functions, recently established for linear stochastic single degree of freedom oscillators, is extended to general finite element systems such as statically indeterminate beam/frame structures and plane stress problems, leading to closed form integral expressions for their dynamic mean and variability response. The integrand of these integral expressions involves the spectral density function of the uncertain material properties and the so called dynamic mean and variability response functions respectively, which are assumed to be deterministic, i.e. independent of the power spectrum as well as the marginal pdf of the uncertain parameters. A finite element method-based fast Monte Carlo simulation procedure is used for the accurate and efficient numerical evaluation of these functions. In order to demonstrate the validity of the proposed procedure, the results obtained using the aforementioned integral expressions are compared to brute-force Monte Carlo simulation. As a further validation of the assumption of independence of the variability response function to the stochastic parameters of the problem, the concept of the generalized variability response function was applied and compared to the steady state dynamic variability response function. The methodology is applied in a dynamically loaded statically indeterminate beam/frame structure and a plane stress problem. The dynamic mean and variability response functions, once established, can be used to perform sensitivity/parametric analyses with respect to various probabilistic characteristics involved in the problem (i.e., correlation distance, standard deviation) and to establish realizable upper bounds on the dynamic mean and variance of the response, at practically no additional computational cost.

Mesh-dependent stability for finite element approximations of parabolic equations with mass lumping

In this paper, based on some mesh-dependent estimates on the extreme eigenvalues of a general finite element system defined on a simplicial mesh, novel and sharp bounds on the permissible time step size are derived for the mass lumping finite element approximations of parabolic equations. The bounds are dependent not only on the mesh size but also on the mesh shape. These results provide guidance to the stability of numerical solutions of parabolic problems in relation to the unstructured geometric meshing. Numerical experiments on both uniform meshes and adaptive meshes are presented to validate the theoretical analysis.

Simple iteration method for structural static reanalysis

This paper presents a simple iterative method for structural static reanalysis. A preconditioned Richardson’s iterative method is developed and the relaxation parameter is determined by a very simple formula derived from the corresponding potential energy function. Based on the iteration method, an acceleration technique is also established. The proposed method is intended to utilize the existing stiffness matrix factorization completed for the initial design, require matrix-by-vector products only, preserve the ease of implementation and improve significantly the quality of the results. The method is suitable for general finite element systems. Calculation of derivatives is not required. The computational time is considerably reduced. A numerical example is used to validate the effectiveness of the proposed reanalysis method.

Efficient Simulation of Coupled Circuit-Field Problems: Generalized Falk Method

The proposed generalized Falk (GF) method offers an extremely simple and convenient way to solve coupled circuit-field problems in circuit simulators by transforming the discretized governing-field equations into guaranteed stable-and-passive one-dimensional (1D) equivalent-circuit systems, which are then combined with the circuit part of the overall coupled problem. More efficient than the traditional Lanczos-type methods, the GF method transforms a general finite-element system represented by a system of full matrices into an identity capacitance (mass) matrix and a tridiagonal conductance (stiffness) matrix. No positive poles are produced; all transformed matrices remain positive definite. The resulting 1D equivalent-circuit system contains only resistors, capacitors, inductors, and current sources. Several numerical examples are provided.

A finite element algorithm for reanalysis of structures with added degrees of freedom

This paper is focused on the reanalysis of structures with added degrees of freedom. By suitably partitioning the extended stiffness matrix, we present a preconditioned conjugate gradient algorithm to analyze the modified structure. The algorithm preserves the ease of implementation and improves significantly the quality of results. In particular, the algorithm can adaptively monitor the accuracy of approximate solutions. It is applicable to general finite element systems. The computational cost is significantly reduced. Numerical examples are given to show the effectiveness of the presented reanalysis algorithm.

A unified reanalysis approach for structural analysis, design, and optimization

This study presents a unified reanalysis approach for structural analysis, design, and optimization that is based on the Combined Approximations (CA) method. The method is suitable for various analysis models (linear, nonlinear, elastic, plastic, static, dynamic), different types of structures (trusses, frames, grillages, continuum structures), and all types of design variables (cross-sectional, material, geometrical, topological). The calculations are based on results of a single exact analysis. The computational effort is usually much smaller than that needed to carry out a complete analysis of modified designs. Accurate results are achieved by low-order approximations for significant changes in the design. It is possible to improve the accuracy by considering higher-order terms, and exact solutions can be achieved in certain cases. The solution steps are straightforward, and the computational procedures presented can readily be used with general finite element systems. Typical results are demonstrated by numerical examples.

Structural modal reanalysis for topological modifications of finite element systems

In this paper, a procedure for modal reanalysis of structures subjected to topological modifications is presented. The procedure is based on results of initial modal analysis of the original structure. The topological modifications include addition of joints, where the number of degrees of freedom (DOF) is increased. The method deals with the stiffness and mass matrices of structures directly, so it can be used with a general finite element system. It is shown that the proposed method is very effective, highly accurate and easy to implement on a computer.

Three-Dimensional Modeling of Damage in Laminated Composites Containing a Central Hole

Renard et all's transverse matrix cracking model is extended to the three-dimensional analysis of laminated composite structures. In this model, a thermodynamic approach to transverse matrix cracking is used. In order to determine the material parameters which modulate the stiffness reduction following the exponential laws assumed by Renard et al., an identification method based on the self-consistent model is proposed. Fiber breakage is detected by the maximum longitudinal strain criterion. The three-dimensional formulation of the damage model taking into account transverse matrix cracking and fiber breakage is essential in order to implement it in the general finite element system, SAMCEF, by which a three-dimensional modeling permits to fully describe the progressive damage in laminated composites containing stress concentrations. This damage model is used to determine the damage accumulations in a laminated graphite/epoxy AS4-3502 plate containing a central hole and under axial tension. Our results are compared to Tan's.

Improved stiffness-based first-order approximations for structural optimization

Improved first-order approximations of displacements, stresses, and forces are presented. The main objectives in developing the method presented are 1) to preserve the ease of implementation and the efficiency of the common first-order approximations and 2) to improve significantly the quality of the results, such that the method can be used in problems with very large changes in the design variables, including geometrical changes and elimination of members. The method is based on results of a single exact analysis and can be used with a general finite element system. It is suitable for different types of design variables and structures. Results obtained by the proposed method are compared with various first-order approximations for modifications in the cross section as well as the geometry and the topology of the structure. It is shown that the proposed approximations are most effective in terms of the accuracy, the efficiency, and the ease of implementation.

A global secant relaxation (GSR) method-based predictor-corrector procedure for the iterative solution of finite element systems

In this paper, an iterative procedure based on improving the diagonal stiffness prediction is used to solve general finite element systems. The procedure consists of a predictor and a corrector, for each iteration step, iteratively obtaining the converged solution. The diagonal stiffness prediction works to predict an incremental response vector for the discrete algebraic system, while the global secant relaxation (GSR) technique works as a corrector, in which a scaling factor is used to adjust the incremental response vector. Drastic reduction of the computer memory requirement can be expected by adopting the diagonal stiffness prediction, and the numerical stability and convergence rate can be improved significantly through the introduction of GSR correction. An efficient and reliable scaling factor is used with which a higher order convergence rate can be expected for the iterative computation.

Efficient sensitivity analysis for structural optimization

Abstract Efficient sensitivity analysis, based on high quality approximations of displacements, is presented. The improved approximations are achieved by combining the efficiency of local approximations (series expansion) and the quality of global approximations (the reduced basis method). Three approximate sensitivity analysis methods are developed and compared: the direct method, the adjoint-variable method and the finite-difference method. All the approximations are based on the results of a single exact analysis of the initial design, and they can be used with a general finite element system. The methods are easy to implement and suitable for different types of design variables and structures. It is shown that high quality approximations of derivatives can be achieved with a relatively small computational effort for very large changes in the design variables.

Plasticity Model for Transversely Isotropic Materials

The most frequently used approach for representing the constitutive relations for anisotropic materials is Hill's incremental plasticity model. However, a fundamental difficulty with the use of Hill's plasticity model is the need to select a unique effective stress‐effective strain relation when none truly exists. Recently, an alternative model based upon stress invariants that does not require definition of an effective stress‐effective strain relation has been proposed. In this study, the invariant‐based model and Hill's model are implemented in a general finite element system. The invariant‐based model is reviewed and a modification is proposed that improves performance for certain multiaxial stress states. Performance of the modified invariant‐based model relative to Hill's model is measured. Comparisons are based upon finite element micromechanics analysis and experimental test results. The invariant‐based model is shown to be superior to Hill's model for a variety of uniaxial and multiaxial loading ...

Experience in the application of a finite element system to the analysis of complex prestressed concrete pressure vessels

The increasing geometric complexity of prestressed concrete pressure vessels (PCPVs) requires an accurate mathematical idealization of the shape of the structure. Finite elements of the isoparametric type provide one answer to this problem. The commercial availability of finite element analysis packages requires the designer to make a careful selection of a suitable package and adapt and extent that system to suit his particular needs. This paper discusses the application by Taywood Engineering Ltd, of the Atkins Stress Analysis System (ASAS) to the analysis of complex, multi-podded PCPVs. Details are given of the modelling process and the choice of grid fineness to be used. The preparation of data and the associated comprehensive checking procedures are described. The selection of material properties is of prime importance in finite element analysis and the dependence of the properties on temperature and creep is discussed together with the way in which results from only three analyses can be combined to provide the designer with the required information. Considerable emphasis is laid on the provision of small pre- and post-processing programs designed to remove the tedium of such large analyses and to leave the designer free to concentrate on the important engineering aspects of his design. The significance of the work lies in the fact that a suite of programs has been built around a general finite element system to satisfy the precise needs of a vessel designer. A design office tool has been produced which is economic in terms of the time and effort required in data preparation and assessment of results.

On the 𝑙² convergence of an algorithm for solving finite element equations

An iterative method of multiple grid type is proposed for solving general finite element systems. It is proved that the method can produce a solution to the equations in O ( N ) O(N) arithmetical operations where N is the number of unknowns.