On the equivalence between the cell-based smoothed finite element method and the virtual element method

We revisit the cell-based smoothed finite element method (SFEM) for quadrilateral elements and extend it to arbitrary polygons and polyhedrons in 2D and 3D, respectively. We highlight the similarity between the SFEM and the virtual element method (VEM). Based on the VEM, we propose a new stabilization approach to the SFEM when applied to arbitrary polygons and polyhedrons. The accuracy and the convergence properties of the SFEM are studied with a few benchmark problems in 2D and 3D linear elasticity. Later, the SFEM is combined with the scaled boundary finite element method to problems involving singularity within the framework of the linear elastic fracture mechanics in 2D

Curvilinear Interface Methodology for Finite-Element Applications

Recent trends in design and manufacturing suggest a tendency toward multiple centers of specialty which results in a need for improved integration methodology for dissimilar finite element or CFD meshes. Since a typical finite element or CFD analysis requires about 50% of an engineers effort to be devoted to modeling and input, there is a need to advance the state-of-the-art in modeling, methodology. These two trends indicate a need to for the capability to combine independently-modeled configurations in an automated and robust way without the need for global remodeling. One approach to addressing this need is the development of interfacing methodology which will automatically integrate independently modeled subdomains. The present research included the following objectives: (i) to develop and implement computational methods for automatically remodeling non-coincident finite element models having a pre-defined interface, (ii) to formulate and implement a parametric representation of general space curves and surfaces with a well-defined orientation, and (iii) to demonstrate the computational methodology with representative two- and three-dimensional finite element models. Methodology for automatically remodeling non-coincident subdomains was developed and tested for two- and three-dimensional, independently modeled subdomains. Representative classes of applications have been solved which gave good agreement with reference solutions obtained with conventional methods. The two-dimensional classes of problems solved included flat and curved membranes multiple subdomains having large gaps between the subdomains and general space curves representing an interface for re-modeling the portions of subdomains adjacent to the interface. The three-dimensional classes of problems solved includes multiple three-dimensional subdomains having large three-dimensional gap between previously modeled subdomains. The interface was represented by general surfaces with a well-defined orientation and having curvature in possibly more than one direction. The results demonstrated the re-modeling methodology to be general, flexible in use, highly automated, and robust for a diverse class of problems. The research reported represents an important advancement in the area of automated re-modeling for computational mechanics applications

A virtual element formulation for general element shapes

The virtual element method is a lively field of research, in which considerable progress has been made during the last decade and applied to many problems in physics and engineering. The method allows ansatz function of arbitrary polynomial degree. However, one of the prerequisite of the formulation is that the element edges have to be straight. In the literature there are several new formulations that introduce curved element edges. These virtual elements allow for specific geometrical forms of the course of the curve at the edges. In this contribution a new methodology is proposed that allows to use general mappings for virtual elements which can model arbitrary geometries. © 2020, The Author(s)

Manufacturability and Analysis of Topologically Optimized Continuous Fiber Reinforced Composites

Researchers are unlocking the potential of Continuous Fiber Reinforced Composites for producing components with greater strength-to-weight ratios than state of the art metal alloys and unidirectional composites. The key is the emerging technology of topology optimization and advances in additive manufacturing. Topology optimization can fine tune component geometry and fiber placement all while satisfying stress constraints. However, the technology cannot yet robustly guarantee manufacturability. For this reason, substantial post-processing of an optimized design consisting of manual fiber replacement and subsequent Finite Element Analysis (FEA) is still required. To automate this post-processing in two dimensions, two (2) algorithms were developed. The first one is aimed at filling the space of a topologically optimized component with fibers of prescribed thickness. The objective is to produce flawless fiber paths, meaning no self-intersections, no tight turns, and no overlapping between fibers. It does so by leveraging concepts from elementary geometry and the Signed Distance Function of a topologically optimized domain. The manufacturable fiber paths are represented using Non-Uniform Rational Basis Splines, which can be readily conveyed to a 3D-printer as The second algorithm then calls a meshing routine to spatially discretize the topologically optimized domain. It takes input from the first algorithm to automatically create and append, orientations and material flags to the spatial elements produced by the meshing routine. Finally, it generates output that is then input to FEA software. The software is written in the C-programming language using the PETSc library. A load case is validated against MSC NASTRAN

Automatic mesh generation

The objective of this thesis project is a study of Pre-Processors and development of an Automatic Mesh Generator for Finite Element Analysis. The Mesh Generator developed in this thesis project can create triangular finite elements from the geometric database of Macintosh Applications. The user is required to give the density parameter to the program for mesh generation. The research is limited to Mesh Generators of planar surfaces. Delauny Triangulation method maximizes the minimum angles of a triangle. Watson\u27s Delauny Triangulation method can mesh only the \u27convex hull\u27 of a set of nodes. This algorithm has been modified to create triangular elements in convex and non-convex surfaces. The surfaces can have holes also. A node generation algorithm to place nodes on and inside a geometry has been developed in this thesis project. The mesh generation is very efficient and flexible. Geometric modeling methods have been studied to understand and integrate the Geometric Modeler with the Finite Element Mesh Generator. Expert Systems can be integrated with Finite Element Analysis. This will make Finite Element Method fully automatic. In this thesis project, Expert Systems in Finite Element Analysis are reviewed. Proposals are made for future approach for the integration of the two fields

A New Approach to Automatic Generation of an all Pentagonal Finite Element Mesh for Numerical Computations over Convex Polygonal Domains

A new method is presented for subdividing a large class of solid objects into topologically simple subregionssuitablefor automatic finite element meshing withpentagonalelements. It is known that one can improve the accuracy of the finite element solutionby uniformly refining a triangulation or uniformly refining a quadrangulation.Recently a refinement scheme of pentagonal partition was introduced in [31,32,33]. It is demonstrated that the numerical solutionbased on the pentagonal refinement scheme outperforms the solutions based on the traditional triangulation refinement scheme as well as quadrangulation refinement scheme. It is natural to ask if one can create a hexagonal refinement or general polygonal refinement schemes with a hope to offer even further improvement. It is shown in literature that one cannot refine a hexagon using hexagons of smaller size. In general, one can only refine an n-gon by n-gons of smaller size if n = 5. Furthermore, we introduce a refinement scheme of a generalpolygon based on the pentagon scheme. This paper first presents a pentagonalization (or pentagonal conversion) scheme that can create a pentagonal mesh from any arbitrary mesh structure. We also introduce a pentagonal preservation scheme that can create a pentagonal mesh from any pentagonal mesh

Subdivision and manifold techniques for isogeometric design and analysis of surfaces

Design of surfaces and analysis of partial differential equations defined on them are of great importance in engineering applications, e.g., structural engineering, automotive and aerospace. This thesis focuses on isogeometric design and analysis of surfaces, which aims to integrate engineering design and analysis by using the same representation for both. The unresolved challenge is to develop a desirable surface representation that simultaneously satisfies certain favourable properties on meshes of arbitrary topology around the extraordinary vertices (EVs), i.e., vertices not shared by four quadrilaterals or three triangles. These properties include high continuity (geometric or parametric), optimal convergence in finite element analysis as well as simplicity in terms of implementation. To overcome the challenge, we further develop subdivision and manifold surface modelling techniques, and explore a possible scheme to combine the distinct appealing properties of the two. The unique advantages of the developed techniques have been confirmed with numerical experiments. Subdivision surfaces generate smooth surfaces from coarse control meshes of arbitrary topology by recursive refinement. Around EVs the optimal refinement weights are application-dependent. We first review subdivision-based finite elements. We then proceed to derive the optimal subdivision weights that minimise finite element errors and can be easily incorporated into existing implementations of subdivision schemes to achieve the same accuracy with much coarser meshes in engineering computations. To this end, the eigenstructure of the subdivision matrix is extensively used and a novel local shape decomposition approach is proposed to choose the optimal weights for each EV independently. Manifold-based basis functions are derived by combining differential-geometric manifold techniques with conformal parametrisations and the partition of unity method. This thesis derives novel manifold-based basis functions with arbitrary prescribed smoothness using quasi-conformal maps, enabling us to model and analyse surfaces with sharp features, such as creases and corners. Their practical utility in finite element simulation of hinged or rigidly joined structures is demonstrated with Kirchhoff-Love thin shell examples. We also propose a particular manifold basis reproducing subdivision surfaces away from EVs, i.e., B-splines, providing a way to combine the appealing properties of subdivision (available in industrial software) for design and manifold basis (relatively new) for analysis.Cambridge International Scholarship Scheme (CISS) by Cambridge Trus

Isogeometric dual mortar methods for computational contact mechanics

International audienceIn recent years, isogeometric analysis (IGA) has received great attention in many fields of computational mechanics research. Especially for computational contact mechanics, an exact and smooth surface representation is highly desirable. As a consequence, many well-known finite e lement m ethods a nd a lgorithms f or c ontact m echanics h ave b een t ransferred t o I GA. I n t he present contribution, the so-called dual mortar method is investigated for both contact mechanics and classical domain decomposition using NURBS basis functions. In contrast to standard mortar methods, the use of dual basis functions for the Lagrange multiplier based on the mathematical concept of biorthogonality enables an easy elimination of the additional Lagrange multiplier degrees of freedom from the global system. This condensed system is smaller in size, and no longer of saddle point type but positive definite. A very simple and commonly used element-wise construction of the dual basis functions is directly transferred to the IGA case. The resulting Lagrange multiplier interpolation satisfies discrete inf–sup stability and biorthogonality, however, the reproduction order is limited to one. In the domain decomposition case, this results in a limitation of the spatial convergence order to O(h 3 /2) in the energy norm, whereas for unilateral contact, due to the lower regularity of the solution, optimal convergence rates are still met. Numerical examples are presented that illustrate these theoretical considerations on convergence rates and compare the newly developed isogeometric dual mortar contact formulation with its standard mortar counterpart as well as classical finite elements based on first and second order Lagrange polynomials