This volume of the SEMA SIMAI Springer Series brings together selected contributions presented at the international conference “eXtended Discretization MethodS” (X-DMS), held during September 2015 in Ferrara, Italy. The conference was one of the thematic conferences supported by the European Community in Computational Methods in Applied Sciences (ECCOMAS) and also one of the special interest conferences sponsored by the International Association for Computational Mechanics (IACM). Twelve minisymposia, more than one hundred oral presentations, and plenary lectures given by eminent personalities in the computational mechanics research field contributed to the scientific value of the event
This book gathers selected contributions on emerging research work presented at the International Conference eXtended Discretization MethodS (X-DMS), held in Ferrara in September 2015. It highlights the most relevant advances made at the international level in the context of expanding classical discretization methods, like finite elements, to the numerical analysis of a variety of physical problems. The improvements are intended to achieve higher computational efficiency and to account for special features of the solution directly in the approximation space and/or in the discretization procedure. The methods described include, among others, partition of unity methods (meshfree, XFEM, GFEM), virtual element methods, fictitious domain methods, and special techniques for static and evolving interfaces. The uniting feature of all contributions is the direct link between computational methodologies and their application to different engineering areas.
Gurkan, C.; Sala-Lardies, E.; Kronbichler, M.; Fernandez, S. Journal of scientific computing Vol. 66, num. 3, p. 1313-1333 DOI: 10.1007/s10915-015-0066-8 Data de publicació: 2015-07-06 Article en revista
A strategy for the Hybridizable Discontinous Galerkin (HDG) solution of problems with voids, inclusions or free surfaces is proposed. It is based on an eXtended Finite Element (X-FEM) philosophy with a level-set description of interfaces. Thus, the computational mesh is not required to fit the interface (i.e.~the boundary), simplifying and reducing the cost of mesh generation and, in particular, avoiding continuous remeshing for evolving interfaces. Differently to previous proposals for HDG solution with non-fitting meshes, here the computational mesh covers the domain, avoiding extrapolations, and ensuring the robustness of the method. The local problem at elements not cut by the interface, and the global problem, are discretized as usual in HDG. A modified local problem is considered at elements cut by the interface. At every cut element, an auxiliary trace variable on the boundary is introduced, which is eliminated afterwards using the boundary conditions on the interface, keeping the original unknowns and the structure of the local problem solver. An efficient and robust methodology for numerical integration in cut elements, in the context of high-order approximations, is also proposed. Numerical experiments demonstrate how X-HDG keeps the optimal convergence, superconvergence, and accuracy of HDG, with no need of adapting the computational mesh to the interface boundary.
High-order Discontinous Galerkin methods are nowadays very popular in the CFD community. DG methods
inherit the advantages of Finite Volume methods (stability through numerical fluxes, local conservation, etc)
but they allow for the use of high-order approximations with a straight-forward implementation of p-adaptivity.
Among all DG methods, the novel Hybridizable Discontinuous Galerkin method (HDG ) has proved
outstanding efficiency. The hybridization process allows reducing the degrees of freedom to the nodes in the
element faces (sides in 2D), similarly to static condensation in CFE.
On other hand, the eXtended Finite Element method (X-FEM ) is a clever strategy to treat the discontinuities
arising at interfaces in bimaterial problems. Interfaces are usually represented as the 0-level set of a signed
distance function, the solution is enriched to represent weak or strong discontinuities across interfaces, and
numerical integration is adapted to take care of the discontinuous approximation inside elements.
This work proposes a formulation for the efficient solution of bimaterial problems, based on these two advanced
discretization techniques: the eXtended Hybridizable Discontinuous Galerkin (X-HDG) method. The
X-FEM philosophy is introduced in an HDG formulation. The solution is enriched with Heaviside functions and,
in the case of weak discontinuities, continuity is weakly imposed, emulating the imposition of continuity across
element boundaries in standard HDG.
As the range of phenomena that need to be simulated in engineering practice broadens, the limitations of conventional computational methods have become apparent. There are many problems of industrial and academic interest which cannot be easily treated with classical methods, requiring the development of new advanced discretization techniques
Cai, Q.; Kollmannsberger, S.; Sala-Lardies, E.; Huerta, A.; Rank, E. Computers & mathematics with applications Vol. 66, num. 12, p. 2545-2558 DOI: 10.1016/j.camwa.2013.09.009 Data de publicació: 2014-01 Article en revista
Inthecaseofdominatingconvection,standardBubnov–Galerkinfiniteelementsareknown to deliver oscillating discrete solutions for the convection–diffusion equation. This paper demonstrates that increasing the polynomial degree ( p -extension) limits these artificial numerical oscillations. This is contrary to a widespread notion that an increase of the poly- nomialdegreedestabilizesthediscretesolution. Thistreatisealsoprovidesexplicitexpres- sionsastowhichpolynomialdegreeissufficientlyhightoobtainstablesolutionsforagiven P ´ eclet number at the nodes of a mesh.
In the case of dominating convection, standard Bubnov–Galerkin finite elements are known to deliver oscillating discrete solutions for the convection–diffusion equation. This paper demonstrates that increasing the polynomial degree (p-extension) limits these artificial numerical oscillations. This is contrary to a widespread notion that an increase of the polynomial degree destabilizes the discrete solution. This treatise also provides explicit expressions as to which polynomial degree is sufficiently high to obtain stable solutions for a given Peclet number at the nodes of a mesh.
Solution of multiphase problems shows discontinuities across the material interfaces,
which are usually weak. Using the eXtended Finite Element Method (X-FEM), these
problems can be solved even for meshes that do not match the geometry. The basic idea is to
enrich the interpolation space by means of a ridge function that is able to reproduce the discontinuity
inside the elements. This approach yields excellent results for linear elements, but fails
to be optimal if high-order interpolations are used.
In this work, we propose a formulation that ensures optimal convergence rates for bimaterial
problems. The key idea is to enrich the interpolation using a Heaviside function that allows the
solution to represent polynomials on both sides of the interface and, provided the interface is
accurately approximated, it yields optimal convergence rates. Although the interpolation is
discontinuous, the desired continuity of the solution is imposed modifying the weak form.
Moreover, in order to ensure optimal convergence, an accurate description of the interface
(which also defines an integration rule for the elements cut by the interface) is needed. Here, we
comment on different options that have been successfully used to integrate high-order X-FEM
elements, and describe a general algorithm based on approximating the interface by piecewise
polynomials of the same degree that the interpolation functions.
'As the range of phenomena that need to be simulated in engineering practice broadens, the limitations of conventional computational methods, such as finite elements (FE), finite volumes or finite difference methods, have become apparent. There are many problems of industrial and academic interest which cannot be easily treated with these classical methods. To overcome the limitations of classical methods, several advanced discretization techniques (mesh-free methods, extended/generalized FE or Dicontinuous Galerkin methods) have recently become very popular in the research community. However, despite their high potential and the important effort devoted to them in the last decade, advanced techniques require still very much attention to reach the popularity of conventional techniques for industrial applications. In fact, engineers are usually not trained in these techniques. The purpose of the ITN research project is to advance in the development and analysis of advanced techniques, with special attention to particular industrial applications of interests in the framework of computational mechanics. However, the introduction of new techniques in industry is only possible if industrial researchers have a deep knowledge and confidence on these techniques and are aware of their advantages. The ITN training program is addressed to researchers that, in the future, may be incorporated in industry. It is based on training-through-research with individual research projects, active participation in network activities and a wide offer of specific courses. In present, the network partners have a wide offer of training courses (joint Erasmus Mundus Master of Science in Computational Mechanics, etc). No experienced researchers or visiting professors are considered in this proposal. The dimension of the academic network teams and the scientific production of all of them clearly demonstrate that they are able to carry out the planned training program.'