Abstract
Abstract This paper presents a study of the complex step differentiation method applied to a parameter sensitivity analysis for 3D elastic contact problem. The analysis is performed with the Boundary Element Method (BEM) using discontinuous elements and the Generalized Newton Method with line search (GNMls). A standard BEM implementation is used and the contact restrictions are fulfilled through the augmented Lagrangian method. This methodology in conjunction with the BEM avoids the calculation of the nonlinear derivatives during the solution process, allowing a fast and reliable solution procedure. The parameter sensitivity is evaluated using complex-step differentiation. This well-known method approximates the derivative of a function analogously to the standard finite differences method, with the advantages of being numerically exact and nearly insensitive to the step-size. As an example, a Hertz-type problem is solved and the sensitivity of the contact pressures with respect to the Young Modulus variation is evaluated. The obtained results are compared with analytical and numerical solutions found in the literature.
Highlights
Contact problems are often found in engineering applications
The mesh has a large element size ratio in order to reduce the overall number of degrees of freedom (DOF) without compromising the solution accuracy
This work evaluated the use of complex step method to obtain sensitivities on Boundary Element Method (BEM)-BEM contact problems with friction
Summary
Contact problems are often found in engineering applications. While some cases can be simplified or even assumed to be irrelevant, there are others where the contact itself is the reason of the engineering problem. The Boundary Element Method (BEM) is well-known for its ability to solve contact problems, since its formulation intrinsically treats the displacements and tractions with same order of approximation This enables the direct application of the contact constraints without the need of penalty parameters or Lagrange multipliers. González et al (2008) and Rodríguez-Tembleque and Abascal (2010) treated FEM-BEM coupled problems using the Augmented Lagrangian formulation to avoid some drawbacks of Lagrange multiplier and penalty methods These approaches were based on the work of Alart and Curnier (1991). Among the existing contact treatment techniques, the Augmented Lagrangian is the most adequate in conjunction with BEM because it eliminates the need of trial and error contact state estimations (Rodríguez-Tembleque et al, 2008) This method imposes the contact restrictions exactly and treats the stick case like a standard two-region BEM formulation. The sensitivity of the contact pressure is analyzed as an application example and compared to the associated analytical derivative
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