Dynamic Relaxation Algorithm Research Articles

A new formulation for fictitious mass of the Dynamic Relaxation method with kinetic damping

This paper deals with the stability and convergence rate of the kinetic damping process in Dynamic Relaxation (DR) method. Based on running an incremental analysis, a new relationship is presented for fictitious mass of the kinetic DR algorithm. This formulation is also combined with a transformational form of the Gerschgörin circles theory, proposed here. For numerical verification, some structures from the finite element models are analyzed statically and dynamically and results are compared with the well-known viscous and kinetic DR algorithms. These examples clearly show that the proposed fictitious mass considerably increases the convergence rate of the kinetic DR iterations.

Nonlinear dynamic structural analysis using dynamic relaxation with zero damping

The main idea of the paper is to present a dynamic relaxation algorithm that does not require the damping matrix and velocity terms. The general formulation suggested in this article covers the common DRM as well. In order to verify the ability of the new technique, the obtained static and dynamic solutions are checked with the ones found by the other scheme. When the loads are variable and the behavior of the system is extremely nonlinear, the proposed procedure works efficiently.

A new method of fictitious viscous damping determination for the dynamic relaxation method

This paper develops a new method for calculating the viscous fictitious damping of the dynamic relaxation (DR) method to overcome one of the most crucial difficulties in its application - the low convergence rate. The DR formulation was derived by error minimizations between two successive iterations to deduce an optimum fictitious mass and viscous damping with the aid of the Stodola iterative process. The efficiency of the new method was verified by its application to a wide range of typical structures with strong nonlinearity. The results show that compared to the conventional DR algorithm such as kinetic approach, the new method improves the convergence rate considerably.

Analysis of clustered tensegrity structures using a modified dynamic relaxation algorithm

Tensegrities are spatial, reticulated and lightweight structures that are increasingly investigated as structural solutions for active and deployable structures. Tensegrity systems are composed only of axially loaded elements and this provides opportunities for actuation and deployment through changing element lengths. In cable-based actuation strategies, the deficiency of having to control too many cable elements can be overcome by connecting several cables. However, clustering active cables significantly changes the mechanics of classical tensegrity structures. Challenges emerge for structural analysis, control and actuation. In this paper, a modified dynamic relaxation (DR) algorithm is presented for static analysis and form-finding. The method is extended to accommodate clustered tensegrity structures. The applicability of the modified DR to this type of structure is demonstrated. Furthermore, the performance of the proposed method is compared with that of a transient stiffness method. Results obtained from two numerical examples show that the values predicted by the DR method are in a good agreement with those generated by the transient stiffness method. Finally it is shown that the DR method scales up to larger structures more efficiently.

The dynamic relaxation method using new formulation for fictitious mass and damping

This paper addresses the modified Dynamic Relaxation algorithm, called mdDR by minimizing displacement error between two successive iterations. In the mdDR method, new relationships for fictitious mass and damping are presented. The results obtained from linear and nonlinear structural analysis, either by finite element or finite difference techniques; demonstrate the potential ability of the proposed scheme compared to the conventional DR algorithm. It is shown that the mdDR improves the convergence rate of Dynamic Relaxation method without any additional calculations, so that, the cost and computational time are decreased. Simplicity, high efficiency and automatic operations are the main merits of the proposed technique.

Design of nexorades or reciprocal frame systems with the dynamic relaxation method

The objective of this paper is to propose a simple method using dynamic relaxation algorithm to find the form and perform a structural analysis of reciprocal frame systems, also called nexorades. After a brief historical note, details of the parameters that govern the design of nexorades are introduced together with details of the assembly. These parameters induce strong geometrical constrains on this kind of structures so that a form-finding step is required. It is suggested here to introduce a fictitious mechanical behaviour to solve this problem. The dynamic relaxation algorithm is used with a model that takes into account the eccentricity between the elements. Its implementation is explained and its versatility is illustrated through several examples covering various fields of applications going from form-finding problems to non-linear structural analysis of structures. In a last example the structural behaviour of nexorades is compared with more conventional triangulated structures.

The effect of aspect ratio on the elastoplastic response of stiffened plates loaded in uniaxial edge compression

A numerical study of the effect of aspect ratio on the buckling and collapse behaviour of flatbar-stiffened plates in compression is described. The plate equations are expressed in finite difference form and the solution is obtained using a dynamic relaxation algorithm. Initial imperfections and residual stresses are also introduced in the study. It was found that, for aspect ratios between 2 and 3.5, buckling occurs in the elastic regime, with the panels exhibiting significant post-buckling strength prior to collapse. For aspect ratios between 1 and 2 buckling occurs as the plastic zone increases, followed by rapid unloading as the panel collapses suddenly. The ultimate strength of the panel reduces with increasing aspect ratio, remaining practically constant at higher aspect ratios. The latter is attributed to the initial single half-wave distortion profile that prohibits the formation of the preferred buckling mode.

Elasto-plastic response of uniformly loaded sector plates: full-section yield model predictions and spread of plasticity

The governing equations for an elasto-plastic large deflection analysis of pressure loaded sector plates, based on the Ilyushin full-section yield criterion and the flow theory of plasticity, are presented. An outline of the solution of these equations using a finite difference implementation of the dynamic relaxation algorithm is given. Numerical results are presented for uniformly loaded slender and stocky sector plates with a 60° sector angle. In particular, first yield data are presented as well as the results of two parameter studies. The first study shows the effects of flexural and in-plane edge conditions on the deflections and stress resultants/couples at the centre of the sector plate and the second contrasts the spread of plasticity in slender sector plates with simply supported and clamped in-plane fixed edges.

Large deflection analysis of eccentrically stiffened sector plates

A new theory for the non-axisymmetric elastic large deflection analysis of sector plates stiffened by a single eccentric rectangular cross-section radial stiffener is presented. In the theoretical model the reaction of the stiffener on the plate is modelled partly by means of body forces in the plate equilibrium equations and by local modification to the loading adjacent to the stiffener. The reaction of the plate on the stiffener is modelled as a distributed force system applied along the length of the stiffener. The force interaction is complemented with a set of plate-stiffener displacement compatibility equations. The governing stiffened plate equations are then transformed into initial value format and discretised with respect to four uniform, interlacing finite-difference meshes. The discrete system of equations is solved using the DR (Dynamic Relaxation) algorithm. The validity and accuracy of the new plate theory is established by comparing large deflection analysis results for a uniformly loaded clamped stiffened sector plate with corresponding results obtained with the ANSYS finite element software. The career analysis is then used to conduct three large deflection parameter studies of simply supported and clamped sector plates stiffened by a single eccentric radial stiffener. Numerical results are presented to show the effect on deflections, stress resultants and stress couples of stiffener depth, plate thickness ratio and sector angle.

Circular plates with one diametral stiffener—An elastic large deflection analysis

A new discretely stiffened circular plate theory is presented in outline. The governing plate equations are solved using a finite-difference implementation of the dynamic relaxation (DR) algorithm. Large deflection numerical solutions are presented for uniformly loaded clamped plates stiffened by a single eccentric rectangular cross-section diametral stiffener. Deflection, stress resultant and stress couple results for two stiffener depths are compared with corresponding results computed with the ANSYS finite element program. It is shown that the two analyses produce results which are not wholly in agreement, particularly in the case of the stress resultants at the centre of the plate. It is suggested that extrapolation and degeneracy procedures used in the finite element analysis may possibly account for the differences in the stress resultants.

Nonlinear first yield analysis of circular plates subjected to combined uniform pressure and edge compression

The large deflection axisymmetric circular plate equations are solved using a finite-difference implementation of the dynamic relaxation (DR) algorithm. The numerical solution is combined with the von Mises yield criterion to provide first yield data for the case of uniform in-plane compression combined with uniform transverse pressure. Data in the form of nondimensional interaction curves are provided for simply supported and clamped plates and for a range of initial deflection amplitudes and slenderness ratios.

Optimal structural membranes—II. Form-finding of prestressed membranes using a curved quadrilateral finite element for surface definition

Form-finding of minimal surface membranes is investigated in this paper. A curved quadrilateral finite element (formulated in Part I) is used to provide a numerical representation of a thin surface (structural membrane) established between fixed or flexible boundaries. Pre-stress is introduced to generate the form. Application of the matrix-based finite element method to the vector-based dynamic relaxation algorithm is presented. When analysing minimal surfaces, the assumption of large strains is shown to lead to a stress deviation at equilibrium. Various techniques are proposed to improve the numerical stability of the solution algorithm. The resulting final numerical model adequately represents both large displacements and large summative strains. Comparisons between numerical and experimental solutions to several minimal surfaces demonstrate the accuracy of the proposed formulation.

NON-LINEAR STRUCTURAL RESPONSE USING ADAPTIVE DYNAMIC RELAXATION ON A MASSIVELY PARALLEL-PROCESSING SYSTEM

A parallel adaptive dynamic relaxation (ADR) algorithm has been developed for nonlinear structural analysis. This algorithm has minimal memory requirements, is easily parallelizable and scalable to many processors, and is generally very reliable and efficient for highly nonlinear problems. Performance evaluations on single-processor computers have shown that the ADR algorithm is reliable and highly vectorizable, and that it is competitive with direct solution methods for the highly nonlinear problems considered. The present algorithm is implemented on the 512-processor Intel Touchstone DELTA system at Caltech, and it is designed to minimize the extent and frequency of interprocessor communication. The algorithm has been used to solve for the nonlinear static response of two and three dimensional hyperelastic systems involving contact. Impressive relative speedups have been achieved and demonstrate the high scalability of the ADR algorithm. For the class of problems addressed, the ADR algorithm represents a very promising approach for parallel-vector processing.

NON‐LINEAR STRUCTURAL RESPONSE USING ADAPTIVE DYNAMIC RELAXATION ON A MASSIVELY PARALLEL‐PROCESSING SYSTEM

A parallel adaptive dynamic relaxation (ADR) algorithm has been developed for non-linear structural analysis. This algorithm has minimal memory requirements, is easily parallelizable and scalable to many processors, and is generally very reliable and efficient for highly non-linear problems. Performance evaluations on single-processor computers have shown that the ADR algorithm is reliable and highly vectorizable, and that it is competitive with direct solution methods for the highly non-linear problems considered. The present algorithm is implemented on the 512-processor Intel Touchstone DELTA system at Caltech, and it is designed to minimize the extent and frequency of interprocessor communication. The algorithm has been used to solve for the non-linear static response of two- and three-dimensional hyperelastic systems involving contact. Impressive relative speed-ups have been achieved and demonstrate the high scalability of the ADR algorithm. For the class of problems addressed, the ADR algorithm represents a very promising approach for parallel-vector processing.

Adaptive dynamic relaxation algorithm for non-linear hyperelastic structures Part II. Single-processor implementation

An efficient single-processor implementation of the Adaptive Dynamic Relaxation (ADR) algorithm is developed. It is designed to exploit data locality and can exploit vectorization of the finite element computations. The ADR algorithm is used to solve for the non-linear static response of two- and three-dimensional hyperelastic systems involving frictionless contact. Performance is compared with an existing finite element code which utilizes a direct solution method. ADR is found to be reliable and highly vectorizable, and it outperforms the direct solution method for the highly non-linear problems considered. In addition, it permits the use of a very simple and efficient contact algorithm. In contrast to direct solution methods, ADR has minimal memory requirements and is easily parallelizable and scalable to more processors. For the class of problems addressed, it represents a very promising approach for parallel-vector processing.

Adaptive dynamic relaxation algorithm for non-linear hyperelastic structures Part III. Parallel implementation

An efficient parallel implementation of the Adaptive Dynamic Relaxation (ADR) algorithm is developed. It is designed to minimize the extent and frequency of interprocessor communication. This version is implemented on a 128-processor Intel hypercube and on a cluster of 16 engineering workstations using PVM. Relative speedups are presented for the non-linear static analysis of two- and three-dimensional hyperelastic systems involving frictionless contact. Impressive speedups are achieved on the hypercube and demonstrate the high scalability of the ADR algorithm. Good speedups are also achieved using PVM, suggesting that PVM is a viable parallel-processing resource for this application.

Adaptive dynamic relaxation algorithm for non-linear hyperelastic structures Part I. Formulation

The formulation of an Adaptive Dynamic Relaxation algorithm with application to non-linear hyperelastic structures is presented. A complete derivation of the Dynamic Relaxation method is given, and the adaptive scheme used to ensure reliability and improve performance is described. Finite element equations are derived for the non-linear analysis of elastic and hyperelastic solids subject to large deformations. The hyperelastic material is assumed to be nearly incompressible, and it is modeled using the Mooney-Rivlin material law using a penalty method to enforce incompressibility. A very simple and efficient contact algorithm based on solver constraints is developed to enforce frictionless contact conditions.

Advances in Dynamic Relaxation Techniques for Nonlinear Finite Element Analysis

Abstract Traditionally, the finite element technique has been applied to static and steady-state problems using implicit methods. When nonlinearities exist, equilibrium iterations must be performed using Newton-Raphson or quasi-Newton techniques at each load level. In the presence of complex geometry, nonlinear material behavior, and large relative sliding of material interfaces, solutions using implicit methods often become intractable. A dynamic relaxation algorithm is developed for inclusion in finite element codes. The explicit nature of the method avoids large computer memory requirements and makes possible the solution of large-scale problems. The method described approaches the steady-state solution with no overshoot, a problem which has plagued researchers in the past. The method is included in a general nonlinear finite element code. A description of the method along with a number of new applications involving geometric and material nonlinearities are presented.

Elastic large deflection response of annular sector plates—a comparison of dr finite-difference, finite element and other numerical solutions

A finite-difference implementation of the dynamic relaxation (DR) algorithm has been used to solve the non-axisymmetric elastic large deflection annular sector plate equations. The equations are introduced without derivation, though some details of the DR algorithm and the finite-difference discretization are presented. The computer software developed has been validated by comparison with small and large deflection solutions reported elsewhere. The small deflection comparison results and some of the large deflection comparisons are presented in tabular format. In addition, a series of graphical comparisons are presented of DR large deflection results with corresponding results obtained from the ANSYS finite element stress analysis package and with other finite element results reported in the literature. In the majority of cases the agreement between the DR and finite element results is very satisfactory.

DR large deflection analysis of sector plates

An elastic large deflection analysis of uniformly loaded sector plates is presented. Numerical solutions are obtained by means of the Dynamic Relaxation (DR) algorithm coupled with an interlacing central finite-difference discretization scheme. Comprehensive verification of the DR results is provided via comparison with exact small deflection solutions and approximate large deflection solutions—the latter being derived from the ANSYS finite element program. Some new large deflection results, in the form of centre deflection vs pressure response curves, are presented for simply supported and clamped plates for a range of sector angles.