P-Adaptive Boundary Elements

This paper presents the implementation of an adaptive philosophy to plane potential problems, using the direct boundary element method. After some considerations about the state of the art and a discussion of the standard approach features, the possibility of separately treating the modelling of variables and their interpolation through hierarchical shape functions is analysed. Then the proposed indicators and estimators are given, followed by a description of a small computer program written for an IBM PC. Finally, some examples show the kind of results to be expected

Original stopping criteria associated tomultilevel adaptive mesh refinement to dealwith local singularities

International audienceThis paper introduces a local multilevel mesh refinement strat-egy that automatically stops relating to a user-defined tolerance even incase of local singular solutions. Refinement levels are automatically gener-ated thanks to a criterion based on the direct comparison of the a posteriorierror estimate with the prescribed error. Singular solutions locally increase with the mesh step (e.g. load discontinuities, point load or geometric in-duced singularities) and are hence characterized by locally large element-wise error whatever the mesh refinement. Then, the refinement criterionmay not be self-sufficient to stop the refinement process. Additional stop-ping criteria are required to avoid an infinite refinement process while stillrespecting the desired threshold. Two original geometry-based stopping cri-teria are proposed that consist in determining the critical region for whichthe mesh refinement becomes inefficient. Numerical examples show the effi-ciency of the methodology for stress tensor approximation in L 2 -relative orL et8734; -absolute norms

A-posteriori error estimates for linear exterior problems via mixed-FEM and DTN mappings

In this paper we combine the dual-mixed finite element method with a Dirichlet-to-Neumann mapping (given in terms of a boundary integral operator) to solve linear exterior transmission problems in the plane. As a model we consider a second order elliptic equation in divergence form coupled with the Laplace equation in the exterior unbounded region. We show that the resulting mixed variational formulation and an associated discrete scheme using Raviart-Thomas spaces are well posed, and derive the usual Cea error estimate and the corresponding rate of convergence. In addition, we develop two different a-posteriori error analyses yielding explicit residual and implicit Bank-Weiser type reliable estimates, respectively. Several numerical results illustrate the suitability of these estimators for the adaptive computation of the discrete solutions

Elastostatics

In this chapter, we are going to describe the main features as well as the basic steps of the Boundary Element Method (BEM) as applied to elastostatic problems and to compare them with other numerical procedures. As we shall show, it is easy to appreciate the adventages of the BEM, but it is also advisable to refrain from a possible unrestrained enthusiasm, as there are also limitations to its usefulness in certain types of problems. The number of these problems, nevertheless, is sufficient to justify the interest and activity that the new procedure has aroused among researchers all over the world. Briefly speaking, the most frequently used version of the BEM as applied to elastostatics works with the fundamental solution, i.e. the singular solution of the governing equations, as an influence function and tries to satisfy the boundary conditions of the problem with the aid of a discretization scheme which consists exclusively of boundary elements. As in other numerical methods, the BEM was developed thanks to the computational possibilities offered by modern computers on totally "classical" basis. That is, the theoretical grounds are based on linear elasticity theory, incorporated long ago into the curricula of most engineering schools. Its delay in gaining popularity is probably due to the enormous momentum with which Finite Element Method (FEM) penetrated the professional and academic media. Nevertheless, the fact that these methods were developed before the BEM has been beneficial because de BEM successfully uses those results and techniques studied in past decades. Some authors even consider the BEM as a particular case of the FEM while others view both methods as special cases of the general weighted residual technique. The first paper usually cited in connection with the BEM as applied to elastostatics is that of Rizzo, even though the works of Jaswon et al., Massonet and Oliveira were published at about the same time, the reason probably being the attractiveness of the "direct" approach over the "indirect" one. The work of Tizzo and the subssequent work of Cruse initiated a fruitful period with applicatons of the direct BEM to problems of elastostacs, elastodynamics, fracture, etc. The next key contribution was that of Lachat and Watson incorporating all the FEM discretization philosophy in what is sometimes called the "second BEM generation". This has no doubt, led directly to the current developments. Among the various researchers who worked on elastostatics by employing the direct BEM, one can additionallly mention Rizzo and Shippy, Cruse et al., Lachat and Watson, Alarcón et al., Brebbia el al, Howell and Doyle, Kuhn and Möhrmann and Patterson and Sheikh, and among those who used the indirect BEM, one can additionally mention Benjumea and Sikarskie, Butterfield, Banerjee et al., Niwa et al., and Altiero and Gavazza. An interesting version of the indirct method, called the Displacement Discontinuity Method (DDM) has been developed by Crounh. A comprehensive study on various special aspects of the elastostatic BEM has been done by Heisse, while review-type articles on the subject have been reported by Watson and Hartmann. At the present time, the method is well established and is being used for the solution of variety of problems in engineering mechanics. Numerous introductory and advanced books have been published as well as research-orientated ones. In this sense, it is worth noting the series of conferences promoted by Brebbia since 1978, wich have provoked a continuous research effort all over the world in relation to the BEM. In the following sections, we shall concentrate on developing the direct BEM as applied to elastostatics

Development of Meshfree Strong-Form Methods

Ph.DDOCTOR OF PHILOSOPH

High-fidelity computational modelling of fluid–structure interaction for moored floating bodies

The development and implementation process of a complete numerical framework for high-fidelity Fluid–Structure Interaction (FSI) simulations of moored floating bodies using Computational Fluid Dynamics (CFD) with the Finite Element Method (FEM) is presented here. For this purpose, the following three main aspects are coupled together: Two-Phase Flow (TPF), Multibody Dynamics (MBD), and mooring dynamics. The fluid–structure problem is two-way and fully partitioned, allowing for high modularity of the coupling and computational efficiency. The Arbitrary Lagrangian–Eulerian (ALE) formulation is used for describing the motion of the mesh-conforming fluid–solid interface, and mesh deformation is achieved with linear elastostatics. Mooring dynamics is performed using gradient deficient Absolute Nodal Coordinate Formulation (ANCF) elements with a two-way mooring–structure coupling and a one-way fluid–mooring coupling. Hydrodynamic loads are applied accurately along mooring cables using the solution of the fluid velocity provided by the TPF solver. For this purpose, fluid mesh elements containing cable nodes that do not conform to the fluid mesh are located with a computationally efficient particle-localisation algorithm. As it is common for partitioned FSI simulations of solids moving within a relatively dense fluid to experience unconditional instability from the added mass effect in CFD, a non-iterative stabilisation scheme is developed here. This is achieved with an accurate and dynamic estimation of the added mass for arbitrarily shaped structures that is then applied as a penalty term to the equations of motion of the solid. It is shown that this stabilisation scheme ensures stability of FSI simulations that are otherwise prone to strong added mass effect without affecting the expected response of structures significantly, even when using fully partitioned fluid–structure coupling schemes. Thorough verification and validation for all aspects of the FSI framework ultimately show that the produced numerical results are in good agreement with experimental data and other inherently stable numerical models, even when complex nonlinear events occur such as vortices forming around sharp corners or extreme wave loads and overtopping on moving structures. It is also shown that the mooring dynamics model can successfully reproduce nonlinearities from high frequency fairlead motions and hydrodynamic loads. The large-scale 3D simulation of a floating semi-submersible structure moored with three catenary lines ties all the models and tools developed here together and shows the capability of the high-fidelity FSI framework to model complex systems robustly and accurately