On the algorithmic and implementational aspects of a Discontinuous Galerkin method at finite strains. (September 2015)

• Record Type: Journal Article
• Authors: Truster, Timothy J.; Chen, Pinlei; Masud, Arif
• Is Part Of: Computers & mathematics with applications. Volume 70:issue 6(2015)
• Page Start: 1266
• Page End: 1289
• Publication Date: 2015-09
• Subjects: Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2015.06.035
• Languages: English
• ISSNs: 0898-1221

Abstract

<abstract xml:lang="en" abstract-type="author" id="a000005"> <title id="st000005">Abstract</title> <sec> <p id="sp000150">In this work, algorithmic modifications are proposed and analyzed for a recently developed stabilized finite strain Discontinuous Galerkin (DG) method. The distinguishing feature of the original method, referred to as VMDG, is a consistently derived expression for the numerical flux and stability tensor that account for evolving material and geometric nonlinearity in the vicinity of the interface. Herein, the proposed modifications involve simplifications to the residual force vector and tangent stiffness matrix of the VMDG method that lead to formulations similar to other existing DG methods but retain the enhanced definition for the stability parameters. The primary objective is to reduce the costs associated with implementing the method as well as executing simulations while retaining accuracy and flexibility, thereby making the formulation amenable to boarder material classes such as inelasticity. Each simplification has associated implications on the mathematical and algorithmic properties of the method, such as <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj2gb2qzff" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si8.gif" display="inline" overflow="scroll" id="d13e3306" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></alternatives></inline-formula> convergence rate, solution accuracy, continuity enforcement, and stability of the nonlinear equation solver. These implications are carefully quantified and assessed through a comprehensive numerical performance study. The range of two and three dimensional problems under consideration involves both isotropic and anisotropic materials. Both triangular and quadrilateral element types are employed along with <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj2gb2rq1p" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si158.gif" display="inline" overflow="scroll" id="d13e3315" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>h</mml:mi></mml:math></alternatives></inline-formula> and <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj2gb2r1w4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si159.gif" display="inline" overflow="scroll" id="d13e3319" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math></alternatives></inline-formula> refinement. The ability of the proposed methods to produce stable and accurate results for such a broad class of problems is highlighted.</p> </sec> </abstract>