Fast Isogeometric Boundary Element Method based on Independent Field Approximation

An isogeometric boundary element method for problems in elasticity is presented, which is based on an independent approximation for the geometry, traction and displacement field. This enables a flexible choice of refinement strategies, permits an efficient evaluation of geometry related information, a mixed collocation scheme which deals with discontinuous tractions along non-smooth boundaries and a significant reduction of the right hand side of the system of equations for common boundary conditions. All these benefits are achieved without any loss of accuracy compared to conventional isogeometric formulations. The system matrices are approximated by means of hierarchical matrices to reduce the computational complexity for large scale analysis. For the required geometrical bisection of the domain, a strategy for the evaluation of bounding boxes containing the supports of NURBS basis functions is presented. The versatility and accuracy of the proposed methodology is demonstrated by convergence studies showing optimal rates and real world examples in two and three dimensions.Comment: 32 pages, 27 figure

Strain gradient continuum mechanics: simplified models, variational formulations and isogeometric analysis with applications

This dissertation is devoted to two families of generalized continuum theories: the first and second strain gradient elasticity theories including the first and second velocity gradient inertia, respectively. First of all, a number of model problems is studied by analytical means revealing the key characters and potential of generalized continuum theories. In particular, the classical Kirsch problem is extended to the case of a simplified first strain gradient elasticity model demonstrating the size dependency of stresses and strains in the vicinity of a round hole in a plate in tension. Within linearly isotropic second strain gradient elasticity theory, instead, a simplified model is proposed, still capable of capturing free surface effects and surface tension, in particular, arising in solids of both nano- and macro-scales. With a series of benchmark problems, including a comprehensive set of stability analyses, the role of higher-order material parameters is revealed. On the way towards computational analysis, the boundary value problems of the fourth- and sixth-order partial differential equations arising in the first and second strain gradient models, respectively, are formulated and analysed in a mathematical variational form within appropriate Sobolev space settings. For numerical simulations, isogeometric Galerkin methods meeting higher-order continuity requirements are implemented in a user element framework of a commercial finite element software. Various benchmarks for statics and free vibrations confirm the optimal convergence properties of the numerical methods, verify the implementation and demonstrate the key properties of the underlying higher-order continuum models. Regarding model validation and applications, thorough analyses of stretching, shearing and vibration phenomena of complex triangular lattices homogenized by the simplified second strain gradient elasticity model reveal the strong size dependency of lattice structures and hence provide pivotal information for practical applications of materials and structures with a microstructure or microarchitecture

Advanced Computational Engineering

The finite element method is the established simulation tool for the numerical solution of partial differential equations in many engineering problems with many mathematical developments such as mixed finite element methods (FEMs) and other nonstandard FEMs like least-squares, nonconforming, and discontinuous Galerkin (dG) FEMs. Various aspects on this plus related topics ranging from order-reduction methods to isogeometric analysis has been discussed amongst the pariticpants form mathematics and engineering for a large range of applications

NURBS-Enhanced Finite Element Method (NEFEM)

The development of NURBS-Enhanced Finite Element Method (NEFEM) is revisited. This technique allows a seamless integration of the CAD boundary representation of the domain and the finite element method (FEM). The importance of the geometrical model in finite element simulations is addressed and the benefits and potential of NEFEM are discussed and compared with respect to other curved finite element techniques.Peer ReviewedPostprint (published version

Computational Gradient Elasticity and Gradient Plasticity with Adaptive Splines

Classical continuum mechanics theories are largely insufficient in capturing size effects observed in many engineering materials: metals, composites, rocks etc. This is attributed to the absence of a length scale that accounts for microstructural effects inherent in these materials. Enriching the classical theories with an internal length scale solves this problem. One way of doing this, in a theoretically sound manner, is introducing higher order gradient terms in the constitutive relations. In elasticity, introducing a length scale removes the singularity observed at crack tips using the classical theory. In plasticity, it eliminates the spurious mesh sensitivity observed in softening and localisation problems by defining the width of the localisation zone thereby maintaining a well-posed boundary value problem. However, this comes at the cost of more demanding solution techniques. Higher-order continuity is usually required for solving gradient-enhanced continuum theories, a requirement difficult to meet using traditional finite elements. Hermitian finite elements, mixed methods and meshless methods have been developed to meet this requirement, however these methods have their drawbacks in terms of efficiency, robustness or implementational convenience. Isogeometric analysis, which exploits spline-based shape functions, naturally incorporates higher-order continuity, in addition to capturing the exact geometry and expediting the design-through-analysis process. Despite its potentials, it is yet to be fully explored for gradient-enhanced continua. Hence, this thesis develops an isogeometric analysis framework for gradient elasticity and gradient plasticity. The linearity of the gradient elasticity formulation has enabled an operator-split approach so that instead of solving the fourth-order partial differential equation monolithically, a set of two partial differential equations is solved in a staggered manner. A detailed convergence analysis is carried out for the original system and the split set using NURBS and T-splines. Suboptimal convergence rates in the monolithic approach and the limitations of the staggered approach are substantiated. Another advantage of the spline-based approach adopted in this work is the ease with which different orders of interpolation can be achieved. This is useful for consistency, and relevant in gradient plasticity where the local (explicit formulation) or nonlocal (implicit formulation) effective plastic strain needs to be discretised in addition to the displacements. Using different orders of interpolation, both formulations are explored in the second-order and a fourth-order implicit gradient formulation is proposed. Results, corroborated by dispersion analysis, show that all considered models give good regularisation with mesh-independent results. Compared with finite element approaches that use Hermitian shape functions for the plastic multiplier or mixed finite element approaches, isogeometric analysis has the distinct advantage that no interpolation of derivatives is required. In localisation problems, numerical accuracy requires the finite element size employed in simulations to be smaller than the internal length scale. Fine meshes are also needed close to regions of geometrical singularities or high gradients. Maintaining a fine mesh globally can incur high computational cost especially when considering large structures. To achieve this efficiently, selective refinement of the mesh is therefore required. In this context, splines need to be adapted to make them analysis-suitable. Thus, an adaptive isogeometric analysis framework is also developed for gradient elasticity and gradient plasticity. The proposed scheme does not require the mesh size to be smaller than the length scale, even during analysis, until a localisation band develops upon which adaptive refinement is performed. Refinement is based on a multi-level mesh with truncated hierarchical basis functions interacting through an inter-level subdivision operator. Through Bezier extraction, truncation of the bases is simplified by way of matrix multiplication, and an element-wise standard finite element data structure is maintained. In sum, a robust computational framework for engineering analysis is established, combining the flexibility, exact geometry representation, and expedited design-through analysis of isogeometric analysis, size-effect capabilities and mesh-objective results of gradient-enhanced continua, the standard convenient data structure of finite element analysis and the improved efficiency of adaptive hierarchical refinement

NURBS-Enhanced Finite Element Method (NEFEM)

The development of NURBS-Enhanced Finite Element Method (NEFEM) is revisited. This technique allows a seamless integration of the CAD boundary representation of the domain and the finite element method (FEM). The importance of the geometrical model in finite element simulations is addressed and the benefits and potential of NEFEM are discussed and compared with respect to other curved finite element techniques