Isogeometric shape optimization in fluid mechanics

Abstract

Document Version Isogeometric Shape Optimization in Fluid MechanicsPeter Nørtoft · Jens Gravesen Abstract The subject of this work is numerical shape optimization in fluid mechanics, based on isogeometric analysis. The generic goal is to design the shape of a 2-dimensional flow domain to minimize some prescribed objective while satisfying given geometric constraints. As part of the design problem, the steady-state, incompressible Navier-Stokes equations, governing a laminar flow in the domain, must be solved. Based on isogeometric analysis, we use B-splines as the basis for both the design optimization and the flow analysis, thereby unifying the models for geometry and analysis, and, at the same time, facilitating a compact representation of complex geometries and smooth approximations of the flow fields. To drive the shape optimization, we use a gradient-based approach, and to avoid inappropriate parametrizations during optimization, we regularize the optimization problem by adding to the objective function a measure of the quality of the boundary parametrization. A detailed description of the methodology is given, and three different numerical examples are considered, through which we investigate the effects of the regularization, of the number of geometric design variables, and of variations in the analysis resolution, initial design and Reynolds number, and thereby demonstrate the robustness of the methodology.

Ship-hull shape optimization with a T-spline based BEM–isogeometric solver

“…Furthermore, Basilevs et al [2] and Scott et al [3] have already demonstrated the efficient usage of IGA in conjunction with T-splines technology [4,5]. Shape optimization, in the context of IGA, has been presented in various works as, e.g., in [6,7,8] for the 2D case and in [9] for the 3D case. In these works, the control points are directly used as shape optimization parameters.…”

Shape-optimization of 2D hydrofoils using an Isogeometric BEM solver

“…There exists, however, works in pertinent literature, e.g., Morino & Bernardini (2001) [38], where P T E is permitted to be a collocation point in the context of a an enhanced formulation that involves an additional integral equation resulting from taking the derivative of the potential Φ(P), P ∈ Ω, with respect to the normal of the wake and letting P → P T E . After solving (6) or (10), we can calculate the hydrofoil's pressure coefficient c p using Bernoulli's equation:…”

“…Moreover, the idea of bridging the gap between Computer-Aided Design (CAD) and Analysis by the introduction of IsoGeometric Analysis (IGA) by Hughes et al (2005) [1], see also Cottrell et al (2009) [2], which directly uses analysis suitable geometric models from the CAD representation, can be efficiently exploited, especially in problems of shape optimization. Shape optimization in the context of IGA has been presented in various works, as e.g., Wall et al (2008) [3], Nagy et al (2009) [4], Nguyen et al (2012) [5], Nortoft & Gravesen (2013) [6], Qian & Sigmund (2011) [7], Cho & Ha (2009) [8], Lian et al (2013) [9], Qian (2010) [10], Li & Qian (2011) [11], where NURBS control points and/or weights have been used as design variables to control the boundary shape. On the other hand, in Kostas et al (2015) [12] and [13], due to the shape complexity of the ship hull, a geometric parametric model using high-level parameters with a direct design meaning has been developed and used for optimization purposes in the context of IGA approach.…”

Planar Parametrization in Isogeometric Analysis

“…In the papers [22,23,26] this method has been successfully applied to 2D shape optimization problems. The Winslow functional is minimized in Step 1 and the linearized Winslow functional is used in Step 2, except for [22] where quasi conformal deformation was used.…”

“…[11,22,23,26]. Every time the geometry changes, that is, at every shape optimization iteration, one needs to update the parametrization in order to maintain the accuracy of the numerical approximation to the PDEs, governing the underlying physical model of the system.…”