Very High Order Anisotropic Metric-Based Mesh Adaptation in 3D. (2016)

• Record Type: Journal Article
• Authors: Coulaud, Olivier; Loseille, Adrien
• Is Part Of: Procedia engineering. Volume 163(2016)
• Page Start: 353
• Page End: 365
• Publication Date: 2016
• Subjects: Metric-based Mesh Adaptation -- High-order interpolation error -- Log-Euclidean framework -- Log-simplex algorithm -- Pk mesh adaptation; Engineering -- Congresses; Engineering -- Periodicals; Engineering; Conference proceedings; Periodicals; 620.005
• DOI: 10.1016/j.proeng.2016.11.071
• Languages: English
• ISSNs: 1877-7058

Abstract

Abstract: In this paper, we study the extension of anisotropic metric-based mesh adaptation to the case of very high-order solutions in 3D. This work is based on an extension of the continuous mesh framework and multi-scale mesh adaptation[10] where the optimal metric is derived through a calculus of variation. Based on classical high order a priori error estimates[4], the point-wise leading term of the local error is a homogeneous polynomial of order k + 1. To derive the leading anisotropic direction and orientations, this polynomial is approximated by a quadratic positive definite form, taken to the power . From a geometric point of view, this problem is equivalent to finding a maximal volume ellipsoid included in the level set one of the absolute value of the polynomial. This optimization problem is strongly non-linear both for the functional and the constraints. We first recast the continuous problem in a discrete setting in the metric-logarithm space. With this approximation, this problem becomes linear and is solved with the simplex algorithm[5] . This optimal quadratic form in the Euclidean space is then found by iteratively solving a sequence of such log-simplex problems. From the field of the local quadratic forms that representing the high-order error, a calculus of variation is used to globally control the error inL p norm. A closed form of the optimal metric is then found. Anisotropic meshes are then generated with this metric based on the unit mesh concept[8] . For the numerical experiments, we consider several analytical functions in 3D. Convergence rate and optimality of the meshes are then discussed for interpolation of orders 1 to 5.