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 p-version of the finite element method for three-dimensional thin walled structures with physically non-linear behaviour

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

SUMMARY

Enforcing essential boundary conditions plays a central role in immersed boundary methods. Nitsche's idea has proven to be a reliable concept to satisfy weakly boundary and interface constraints. We formulate an extension of Nitsche's method for elasticity problems in the framework of higher order and higher continuity approximation schemes such as the B-spline and non-uniform rational basis spline version of the finite cell method or isogeometric analysis on trimmed geometries. Furthermore, we illustrate a significant improvement of the flexibility and applicability of this extension in the modeling process of complex 3D geometries. With several benchmark problems, we demonstrate the overall good convergence behavior of the proposed method and its good accuracy. We provide extensive studies on the stability of the method, its influence parameters and numerical properties, and a rearrangement of the numerical integration concept that in many cases reduces the numerical effort by a factor two. A newly composed boundary integration concept further enhances the modeling process and allows a flexible, discretization-independent introduction of boundary conditions. Finally, we present our strategy in the framework of the modeling and isogeometric analysis process of trimmed non-uniform rational basis spline geometries. Copyright © 2013 John Wiley & Sons, Ltd.

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Weakly enforced essential boundary conditions for NURBS-embedded and trimmed NURBS geometries on the basis of the finite cell method

In the p-version of the finite element method, the triangulation is fixed and the degree p, of the piecewise polynomial approximation, is progressively increased until some desired level of precision is reached. In this paper, we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation. These properties lead to an a priori estimate of the asymptotic rate of convergence of the p-version. The estimate shows that the p-version gives results which are not worse than those obtained by the conventional finite element method (called the h-version, in which h represents the maximum diameter of the elements), when quasi-uniform triangulations are employed and the basis for comparison is the number of degrees of freedom. Furthermore, in the case of a singularity problem, we show (under conditions which are usually satisfied in practice) that the rate of convergence of the p-version is twice that of the h-version with quasi-uniform mesh. Inverse approximation theorems which determine the smoothness of a function based on the rate at which it is approximated by piecewise polynomials over a fixed triangulation are proved for both singular and nonsingular problems. We present numerical examples which illustrate the effectiveness of the p-version for a simple one-dimensional problem and for two problems in two-dimensional elasticity.We also discuss roundott error and computational costs associated with the p-version. Finally, we describe some important features, such as hierarchic basis functions, which have been utilized in COMET-X, an experimental computer implemen- tation of the p-version.

The p‐Version of the Finite Element Method

Applying the hp–d version of the FEM to locally enhance dimensionally reduced models

In this paper we present a computer aided method supporting co-operation between different project partners, such as architects and engineers, on the basis of strictly three-dimensional models. The center of our software architecture is a product model, described by the Industry Foundation Classes (IFC) of the International Alliance for Interoperability (IAI). From this a geometrical model is extracted and automatically transferred to a computational model serving as a basis for various simulation tasks. In this paper the focus is set on the advantage of the fully three-dimensional structural analysis performed by p-version of the finite element analysis. Other simulation methods are discussed in a separate contribution of this Volume (Treeck 2004). The validity of this approach will be shown in a complex example.

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Structural Analysis based on the Product Model Standard IFC

A high order finite element approach is applied to elastoplastic problems in two as well as in three dimensions. The element formulations are based on quadrilaterals and hexahedrals, taking advantage of the blending function method in order to accurately represent the geometry. A comparison of h- and p-extensions is drawn and it is shown that thin-walled structures commonly being analysed by dimensionally reduced elements may be consistently discretized by high order hexahedral elements leading to reliable and efficient computations even in case of physically nonlinear problems.

A Numerical Investigation of High-Order Finite Elements for Problems of Elastoplasticity

Dealing with surface-oriented models - e.g. B-Rep models - is very popular and appropriate for many applications. They can be read by most CAD programs and they provide all freedom of modelling. Concerning a lot of other tasks - consistency checks, collision detection, structural analysis, flow simulation, e.g. - these models become difficult to handle, and a volume-oriented model has to be derived from the existing surface-oriented one. Hierarchical volume-oriented models, represented by octrees for example, provide an easy access to solve the latter tasks with respect to their spatial decomposition of the underlying geometry. This paper deals with fast and efficient algorithms to generate and process octrees - even on-the-fly - from surface-oriented models for applications in civil engineering. Encoding these octrees as binary streams makes them suitable to get multiplexed with other octree-coded objects or for the usage in pipe-like constructs. Conventional algorithms for octree generation or processing don't exploit the full potential of these structures. In spite of the principal advantages of octrees concerning complexity, objects of a higher resolution typically still entail too high run-time and memory requirements. Usually, an expensive floating-point-based decision whether or not to refine the structure has to be taken in each voxel (cell) successively. In our approach, instead, the refinement decision is done by a simple parameter comparison of plane equations, avoiding all these costs. By treating each face of the surface-oriented model as a plane that divides the whole space into two half spaces - inside and outside -, the volume-oriented model can be built from intersecting all inside-attributed half spaces. The steps for generating an octree presentation for each corresponding plane, intersecting these octrees, and encoding the result as a binary stream can be done at once - thus, the octree generation is free of any redundant calculations, and the overall memory requirements are reduced to a minimum due to the usage of stacks. The highest gain can be achieved in run-time, e.g. an octree generation for an average geometry with more than 1.5 billion voxels can be done in best time on a standard PC. Several of these binary streams can be multiplexed to perform further Boolean or more sophisticated operations (e.g. collision detection), while one always has the choice to perform this operations on-the-fly or to perform consecutive operations - like with Unix pipes - on binary streams written to the hard disk. One target application of this method deals with consistency checks for CAD models in the scope of simplifying and unifying planning processes in civil engineering. Before a connection model for structural analysis is created out of an (Eurostep) IFC model, any modelling errors (geometric inconsistencies) - wrong intersections or gaps between parts of the model - can be detected fast and easily. Hence, this proceeding enables us to obtain a reliable volume-oriented attributed model that can serve for numerical simulations as well as to determine relations between parts of the model to ensure global consistency, which brings us one step closer to the long-term objective of completely embedded simulation processes.

Efficient Algorithms for Octree-Based Geometric Modelling

In structural mechanics, a large variety of finite element approaches are used, some of them  especially of p-type  without an inherent hierarchical substructuring. This often turns out to be a drawback. By embedding the finite element decomposition into an octree structure, the elements can be arranged in a hierarchical way, which does not only open the door to efficient iterative solvers based on the classical nested dissection algorithm, but also allows to speed up the solution process in case only parts of the underlying geometric model are changed, as only those parts and their region of direct influence have to be recomputed.

A. Niggl

Papers

Applying the hp–d version of the FEM to locally enhance dimensionally reduced models

Structural Analysis based on the Product Model Standard IFC

A Numerical Investigation of High-Order Finite Elements for Problems of Elastoplasticity

Efficient Algorithms for Octree-Based Geometric Modelling

Extending the p -Version of Finite Elements by an Octree-Based Hierarchy