Optimization-Based Strong Coupling Procedure for Partitioned Analysis

Abstract

The concept of partitioning a complex engineering problem into smaller, manageable components and investigating each individual component autonomously has been in use for many decades. Such partitioning approaches, however, rely on strong and occasional unwarranted assumptions regarding the interactions among different engineering components. Fluid and structure interaction, soil and structure interaction, and human and structure interaction are but a few of the many such partitioned analyses commonly needed in civil engineering applications. Recently, there has been a growing interest in combining the expertise developed separately in traditionally distinct fields to obtain a holistic treatment of engineering problems. Such holistic treatment would ultimately yield not only more realistic and accurate analyses of coupled systems but also improved optimality in engineering designs. This growing interest has resulted in development of mathematical coupling procedures for conjoining multiple, separately developed, single-solver numerical models along their interfaces. The present manuscript contributes to the field of partitioned analysis by introducing a novel mathematical coupling procedure based on the minimization of an objective function consisting of coupling conditions. The authors' approach to coupling implements optimization techniques and is observed to eliminate the divergence issues that may be encountered with iterative coupling methods. The proposed optimization-based coupling scheme is compared against the well-known block Gauss-Seidel (BGS) iteration method and considers two aspects: the accuracy of the coupled model predictions and the convergence of the coupled parameters. The comparison is completed for three case studies with increasing complexity: a linear set of equations, polynomials with random coefficients, and a linear dynamic system.

Numerical Integration of Coupled Equations for High-Speed Electromechanical Devices

“…Similarly, the torque and force in (1b) and (1c) depend on the (unknown) positions and velocities. When a strong coupling between the equations is required [10], [11], iterative procedures based on Gauss-Seidel or to Newton-Raphson schemes are usually adopted. They rely on the efficient evaluation (also approximated) of a number of derivatives [11], [12].…”

“…When a strong coupling between the equations is required [10], [11], iterative procedures based on Gauss-Seidel or to Newton-Raphson schemes are usually adopted. They rely on the efficient evaluation (also approximated) of a number of derivatives [11], [12]. These schemes when applied to the problem at hand may require very long computations [10].…”

Prioritization of Code Development Efforts in Partitioned Analysis

“…Finding the correct values for these dependent variables is the main question to be solved in strong coupling problems. Various strong coupling algorithms are proposed in the literature (Figures 2a-d): the Block-Jacobi method (Matthies et al, 2006;Fernandez and Moubachir, 2005), the Block Gauss-Siedel method (Joosten et al, 2009;Matthies et al, 2006), gradient-based Newton-like methods (Heil, 2004;Matthies and Steindorf, 2002b;Matthies and Steindorf, 2003;Fernandez and Moubachir, 2005), and optimization-based methods (Farajpour and Atamturktur, 2012). Note that in partitioned analysis, quantification, propagation, and mitigation of uncertainties in input parameters play an important role for the complete validation of coupled models (Avramova and Ivanov, 2010), which is currently an active research area.…”

Resource allocation for code development in partitioned models

“…Y dep in these equations then represents the combined set of values transferred from one constituent to another. Several methods have been proposed to solve for the coupling conditions (Matthies et al, 2006); for instance, Newton-like coupling methods Steindorf, 2002a, b, 2003;Fernandez and Moubachir, 2005), Block-Jacobi and Block-Gauss-Seidel methods ( Joosten et al, 2009), and optimization-based coupling (OBC) method (Farajpour and Atamturktur, 2012). The OBC method overcomes the divergence problems that classical coupling techniques, such as Block-Gauss-Seidel, may face.…”

“…The OBC method overcomes the divergence problems that classical coupling techniques, such as Block-Gauss-Seidel, may face. Furthermore, the simultaneous execution of constituent models makes OBC suitable for parallel computing (Farajpour and Atamturktur, 2012). Therefore, the OBC is preferable for this work.…”