Timestep Selection for Dynamic Relaxation Method

Rezaiee‐Pajand, Mohammad; Kadkhodayan, Mehran; Alamatian, Javad

doi:10.1080/15397734.2011.599311

Cited by 27 publications

(15 citation statements)

References 38 publications

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“…The iterations are inherently unstable in the standard DRM because the numerical time integration is utilized for solving the differential equations of motion [23][24][25][26]. Hence, the stability of DRM depends on fictitious parameters including the time step, diagonal mass, and damping matrices.…”

Section: Modifications Of the Standard Drmmentioning

confidence: 99%

“…11produces the highest convergence rate. Sometimes, an extremely small or a very large value is obtained for the time step, which may cause numerical instability [25].…”

Section: Optimal Time Stepmentioning

confidence: 99%

“…By minimizing the unbalanced energy function in each iteration, Rezaiee-Pajand et al [25] introduced a new fictitious time step. Based on their study, the residual energy of a structural system can be expressed as a function from the residual force and displacement vectors.…”

Section: Residual Energy Minimizer Time Stepmentioning

confidence: 99%

“…In fact, the explicit elastic and geometric stiffness matrices are provided by using Eqs. (25) and (30) for plane elasticity problems. In the next stage, three-dimensional rotational effects of each element should be considered.…”

Section: The Geometric Stiffness Matrix Of Plane-stress Triangular Elmentioning

confidence: 99%

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Efficiency of dynamic relaxation methods in form-finding of tensile membrane structures

2019

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The main idea of this study is to investigate and compare the efficiency of the dynamic relaxation methods (DRMs) in the form-finding of tensile membrane structures (TMSs). The form-finding process as a main stage in the design and construction of TMSs has been considered for finding an equilibrium configuration subjected to a specific prestress distribution. DRMs as a pseudo-dynamic analysis are an explicit iterative technique for the nonlinear analysis of structures. In the techniques, the static equilibrium of structures is solved by integrating the pseudo-dynamic equations in order to obtain the steady state of the pseudo-dynamic problem. In this study, the efficiency and generality performance of DRMs are compared through solving three selected TMSs. In order to achieve this purpose, seven schemes of the DR approach are selected based on combining the fictitious parameters including the time step, diagonal mass, and damping matrices which were proposed in the previous studies. Furthermore, a reference index is proposed by combining the total number of iterations and the overall duration of analysis in order to appropriately compare the schemes of the DR approach in the form-finding of TMSs.

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Section: Modifications Of the Standard Drmmentioning

confidence: 99%

“…11produces the highest convergence rate. Sometimes, an extremely small or a very large value is obtained for the time step, which may cause numerical instability [25].…”

Section: Optimal Time Stepmentioning

confidence: 99%

Section: Residual Energy Minimizer Time Stepmentioning

confidence: 99%

Section: The Geometric Stiffness Matrix Of Plane-stress Triangular Elmentioning

confidence: 99%

See 2 more Smart Citations

Efficiency of dynamic relaxation methods in form-finding of tensile membrane structures

2019

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“…For instance, minimization of nodal residual forces between two consecutive DR iterations yields the Modified Fictitious Time (MFT) algorithm in which the time increment is updated in each iteration . In addition, Rezaiee‐Pajand et al utilized the out‐of‐balance energy as a criterion to derive new relationship for the fictitious time step . Because of explicit formulation and low storage requirements, DR method is a suitable and efficient alternative to Newton–Raphson (N–R) methods that is commonly used in nonlinear implicit dynamic analysis.…”

Section: Introductionmentioning

confidence: 99%

Dynamic relaxation method based on Lanczos algorithm

Namadchi

Alamatian

2017

Int. J. Numer. Meth. Engng

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This paper tries to accelerate the convergence rate of the general viscous dynamic relaxation method. For this purpose, a new automated procedure for estimating the critical damping factor is developed by employing a simple variant of the Lanczos algorithm, which does not require any re-orthogonalization process. All of the computational operations are performed by simple vector-matrix multiplication without requiring any matrix factorization or inversion. Some numerical examples with geometric nonlinear behavior are analyzed by the proposed algorithm. Results show that the suggested procedure could effectively decrease the total number of convergence iterations compared with the conventional dynamic relaxation algorithms.A. H. NAMADCHI AND J. ALAMATIAN time integration process. Large amount of research has been directed toward both improving the DR parameters and application of the DR method in various engineering problems. Otter employed DR to analyze axially symmetric pressure vessels and a three-dimensional elastic arch dam using central finite differences [5,6]. Cassel investigated the application of DR technique in analyze of cylindrical shells [7]. Rushton proposed a systematic procedure to tackle rectangular plate problems including small and large deflection of elastic plates as well as the buckling behavior [8][9][10]. In order to damp out the artificial oscillations as quick as possible, Rushton suggested that the critical damping of fundamental mode of vibration is selected as the fictitious damping factor, and an automated process for estimating this factor was also introduced [8]. Considering DR as a fictitious time-marching scheme, Brew and Brotton investigated DR characteristics based on the concept of amplification matrix and employed DR to analyze frame structures [11]. Pseudo-transient analysis of linear and geometric nonlinear Mindlin plates using finite element method was successfully carried out by adopting a specific form of DR. Some empirically based suggestions regarding time-step selection and fictitious mass matrix were also reported [12]. By assuming a load vector proportional to diagonal entries of the artificial mass matrix, which is formulated using Gershgorin's circle theorem, Bunce proposed a new technique to determine critical damping in DR algorithm based on the Rayleigh's principle [13]. In another study, Papadrakakis evaluated DR properties by transforming the fundamental difference equations into the error space, which leads to an eigenvalue problem. By examining the resultant characteristic equation, a new automated procedure for estimating DR fictitious parameters was proposed that improves the convergence rate of the method [3]. Zienkiewicz and Löhner studied the concept of viscous relaxation in which the inertial term is removed from the virtual dynamic system. They then utilized a line search technique to accelerate the convergence [14]. Instead of applying the external load in an incremental form, which is conventional in nonlinear quasi-static analyses, a novel techniq...

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Improving the convergence rate of kinetic dynamic relaxation method with new R-point

Alamatian

2021

Engineering with Computers

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Timestep Selection for Dynamic Relaxation Method

Cited by 27 publications

References 38 publications

Efficiency of dynamic relaxation methods in form-finding of tensile membrane structures

Efficiency of dynamic relaxation methods in form-finding of tensile membrane structures

Dynamic relaxation method based on Lanczos algorithm

Improving the convergence rate of kinetic dynamic relaxation method with new R-point

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