Analysis of a finite element formulation for modelling phase separation

In Combescure, A., De Borst, R., and Belytschko, T., editors, IUTAM Symposium on Discretization Methods for Evolving Discontinuities, volume 5 of IUTAM Bookseries, pages 89–102. Springer.The Cahn-Hilliard equation is of importance in materials science and a range of other fields. It represents a diffuse interface model for simulating the evolution of phase separation in solids and fluids, and is a nonlinear fourth-order parabolic equation, which makes its numerical solution particularly challenging. To this end, a finite element formulation has been developed which can solve the Cahn-Hilliard equation in its primal form using C^0 basis functions. Here, analysis of a fully discrete version of this method is presented in the form of a priori uniqueness, stability and error analysis

Embedded discontinuities for softening solids

Additional, discontinuous functions are added to the displacement field of standard finite elements in order to capture highly localised zones of intense straining. By embedding discontinuities within an element it is possible to effectively model localisation phenomena (such as fracture in concrete) with a relatively small number of finite elements. The displacement jump is regularised, producing bounded strains and allowing the application of classical strain softening constitutive laws. It is then possible to achieve mesh-objective results with respect to energy dissipation without resorting to higher-order continuum theories

Discrete analysis of localisation in three-dimensional solids

A procedure is illustrated for the determination of the normal direction of a discontinuity plane within a solid finite element. Using so-called embedded discontinuities, discrete constitutive models can be applied within a continuum framework. A significant difficulty within this method for three-dimensional problems is the determination of the normal direction for a discontinuity. Bifurcation analysis indicates the development of a discontinuity and multiple solution for the normal. The procedure developed here chooses the appropriate normal by exploiting features of the embedded discontinuity method

DOLFIN: Automated Finite Element Computing

We describe here a library aimed at automating the solution of partial differential equations using the finite element method. By employing novel techniques for automated code generation, the library combines a high level of expressiveness with efficient computation. Finite element variational forms may be expressed in near mathematical notation, from which low-level code is automatically generated, compiled and seamlessly integrated with efficient implementations of computational meshes and high-performance linear algebra. Easy-to-use object-oriented interfaces to the library are provided in the form of a C++ library and a Python module. This paper discusses the mathematical abstractions and methods used in the design of the library and its implementation. A number of examples are presented to demonstrate the use of the library in application code

Design With Attitude: A Key World Class Method

World Class Design by implication can best be achieved by adopting world class methods. ICED conferences continue to disseminate methodologies to help make designers and design teams more effective and efficient. At ICED 95 we introduced our PAKTS model (Figure 1.) which identified the total educational needs of young engineering designers [Robotham 1995]. The five key elements of the model are Processes, Attitudes, Knowledge, Tools and Skills. Of these elements, the least developed in terms of methodologies is that of “Attitude”. This paper will give further consideration to this element and identify the qualities that a designer must develop to be world class. These qualities would include: professionalism, ownership, responsibility, team player, discipline, tenacity, self-reliance, and self-learner. We would hope that this paper will form the basis of a “Design with Attitude” method, that would sit alongside all other world class methods, because we firmly believe that World Class Design cannot be achieved without the designer adopting a “Design with Attitude” approach

A study of discontinuous Galerkin methods for thin bending problems

Various continuous/discontinuous Galerkin formulations are examined for the analysis of thin plates. These methods rely on weak imposition of continuity of the normal slope across element boundaries. We draw here upon developments in discontinuous Galerkin methods for second-order elliptic equations, for which several unconditionally stable methods are known, and present continuous/discontinuous Galerkin formulations for bending problems inspired by these methods. For each approach, benchmark simulations have been performed and compared. Also, conclusions are drawn on to the computational ef ciency of the different methods