TY - JOUR

T1 - Practical Integration of Semidiscretized Nonlinear Equations of Motion

T2 - Proper Convergence for Systems with Piecewise Linear Behavior

AU - Soroushian, Aram

AU - Wriggers, Peter

AU - Farjoodi, Jamshid

PY - 2012/3/17

Y1 - 2012/3/17

N2 - Time integration is the most versatile tool for analyzing semidiscretized equations of motion. The responses are approximations, with deviations from the exact responses mainly depending on the integration method and the integration step sizes. When repeating the analyses with smaller steps, the responses generally converge to the exact responses. However, the convergence trends are different in linear and nonlinear analyses.Whereas in linear analyses, by decreasing the sizes of integration steps, the errors decrease with a rate, depending on the orders of accuracy, in nonlinear analyses, the change in errors might be unpredictable. The main reason is the inconsistency between the integration steps sizes and the residuals of nonlinearity iterations. In this paper, based on careful selection of nonlinearity tolerances, a methodology and a method to overcome this inconsistency for semidiscretized systems with piecewise linear behavior are introduced. When the responses converge, except for systems with very complex behaviors, the proposed method leads to proper convergence, with tolerable computational costs. In addition, by implementing the proposed method, more reliable error estimations can be expected from convergence-based accuracy controlling methods.

AB - Time integration is the most versatile tool for analyzing semidiscretized equations of motion. The responses are approximations, with deviations from the exact responses mainly depending on the integration method and the integration step sizes. When repeating the analyses with smaller steps, the responses generally converge to the exact responses. However, the convergence trends are different in linear and nonlinear analyses.Whereas in linear analyses, by decreasing the sizes of integration steps, the errors decrease with a rate, depending on the orders of accuracy, in nonlinear analyses, the change in errors might be unpredictable. The main reason is the inconsistency between the integration steps sizes and the residuals of nonlinearity iterations. In this paper, based on careful selection of nonlinearity tolerances, a methodology and a method to overcome this inconsistency for semidiscretized systems with piecewise linear behavior are introduced. When the responses converge, except for systems with very complex behaviors, the proposed method leads to proper convergence, with tolerable computational costs. In addition, by implementing the proposed method, more reliable error estimations can be expected from convergence-based accuracy controlling methods.

KW - Accuracy-controlling methods

KW - Nonlinearity residual

KW - Nonlinearity tolerance

KW - Order of accuracy

KW - Piecewise linear

KW - Proper convergence

KW - Pseudoerror

KW - Space of nonlinearity

KW - Time integration

KW - Time step

UR - http://www.scopus.com/inward/record.url?scp=84879533845&partnerID=8YFLogxK

U2 - 10.1061/(ASCE)EM.1943-7889.0000434

DO - 10.1061/(ASCE)EM.1943-7889.0000434

M3 - Article

AN - SCOPUS:84879533845

VL - 139

SP - 114

EP - 145

JO - Journal of engineering mechanics

JF - Journal of engineering mechanics

SN - 0733-9399

IS - 2

ER -