高等学校計算数学学報 = Numerical mathematics

https://apps.dtic.mil/sti/pdfs/ADA558914.pdf

An Isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces

The finite cell method is an embedded domain method, which combines the fictitious domain approach with higher-order finite elements, adaptive integration, and weak enforcement of unfitted essential boundary conditions. Its core idea is to use a simple unfitted structured mesh of higher-order basis functions for the approximation of the solution fields, while the geometry is captured by means of adaptive quadrature points. This eliminates the need for boundary conforming meshes that require time-consuming and error-prone mesh generation procedures, and opens the door for a seamless integration of very complex geometric models into finite element analysis. At the same time, the finite cell method achieves full accuracy, i.e. optimal rates of convergence, when the mesh is refined, and exponential rates of convergence, when the polynomial degree is increased. Due to the flexibility of the quadrature based geometry approximation, the finite cell method can operate with almost any geometric model, ranging from boundary representations in computer aided geometric design to voxel representations obtained from medical imaging technologies. In this review article, we first provide a concise introduction to the basics of the finite cell method. We then summarize recent developments of the technology, with particular emphasis on the research topics in which we have been actively involved. These include the finite cell method with B-spline and NURBS basis functions, the treatment of geometric nonlinearities for large deformation analysis, the weak enforcement of boundary and coupling conditions, and local refinement schemes. We illustrate the capabilities and advantages of the finite cell method with several challenging examples, e.g. the image-based analysis of foam-like structures, the patient-specific analysis of a human femur bone, the analysis of volumetric structures based on CAD boundary representations, and the isogeometric treatment of trimmed NURBS surfaces. We conclude our review by briefly discussing some key aspects for the efficient implementation of the finite cell method.

The Finite Cell Method: A Review in the Context of Higher-Order Structural Analysis of CAD and Image-Based Geometric Models

Isogeometric collocation: Cost comparison with Galerkin methods and extension to adaptive hierarchical NURBS discretizations

Weak coupling for isogeometric analysis of non-matching and trimmed multi-patch geometries

We consider a phase field model for the formulation and solution of topology optimization problems in the minimum compliance case. In this model, the optimal topology is obtained as the steady state of the phase transition described by the generalized Cahn–Hilliard equation which naturally embeds the volume constraint on the amount of material available for distribution in the design domain. We reformulate the model as a coupled system and we highlight the dependency of the optimal topologies on dimensionless parameters. We consider Isogeometric Analysis for the spatial approximation which facilitates encapsulating the exactness of the representation of the design domain in the topology optimization and is particularly suitable for the analysis of phase field problems. We demonstrate the validity of the approach and numerical approximation by solving two and three-dimensional topology optimization problems.

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Isogeometric Analysis for Topology Optimization with a Phase Field Model

Isogeometric analysis of the advective Cahn-Hilliard equation: Spinodal decomposition under shear flow

The finite cell method (FCM) belongs to the class of immersed boundary methods, and combines the fictitious domain approach with high-order approximation, adaptive integration and weak imposition of unfitted Dirichlet boundary conditions. Its main idea consists of the extension of the physical domain of interest beyond its potentially complex boundaries into a larger embedding domain of simple geometry, which can be meshed easily by a structured grid. We present an isogeometric design-through-analysis methodology based on the B-spline version of the finite cell method, which allows for the seamless integration of fully three-dimensional parameterizations of complex engineering parts described by T-spline surfaces into finite element analysis. The approach is demonstrated to achieve optimal rates of convergence and to yield accurate stress results not only within the domain of interest, but also directly on the immersed boundary. We also show that hierarchical refinement of B-splines considerably increases the flexibility of the immersed boundary approach in terms of adaptive resolution of local features in the geometry and the solution fields. At the same time, hierarchical refinement maintains the key advantage of fully automated mesh generation for complex geometries due to its simplicity and straightforward implementation. We illustrate the versatility of our methodology by two complex industrial examples of a ship propeller and an automobile wheel.

Micheal J. Borden

Papers

Isogeometric Analysis for Topology Optimization with a Phase Field Model

Isogeometric analysis of the advective Cahn-Hilliard equation: Spinodal decomposition under shear flow

Isogeometric Analysis and the Finite Cell method