Mark A. Walkley

A FINITE ELEMENT METHOD FOR THE ONE-DIMENSIONAL EXTENDED BOUSSINESQ EQUATIONS

SUMMARY A new finite element method for Nwogu’s (O. Nwogu, ASCE J. Waterw., Port, Coast., Ocean Eng., 119, 618‐638 (1993)) one-dimensional extended Boussinesq equations is presented using a linear element spatial discretisation method coupled with a sophisticated adaptive time integration package. The accuracy of the scheme is compared to that of an existing finite difference method (G. Wei and J.T. Kirby, ASCE J. Waterw., Port, Coast., Ocean Eng., 121, 251‐261 (1995)) by considering the truncation error at a node. Numerical tests with solitary and regular waves propagating in variable depth environments are compared with theoretical and experimental data. The accuracy of the results confirms the analytical prediction and shows that the new approach competes well with existing finite difference methods. The finite element formulation is shown to enable the method to be extended to irregular meshes in one dimension and has the potential to allow for extension to the important practical case of unstructured triangular meshes in two dimensions. This latter case is discussed. Copyright

Finite Element Simulation of Three-Dimensional Free-Surface Flow Problems

An adaptive finite element algorithm is described for the stable solution of three-dimensional free-surface-flow problems based primarily on the use of node movement. The algorithm also includes a discrete remeshing procedure which enhances its accuracy and robustness. The spatial discretisation allows an isoparametric piecewise-quadratic approximation of the domain geometry for accurate resolution of the curved free surface. The technique is illustrated through an implementation for surface-tension-dominated viscous flows modelled in terms of the Stokes equations with suitable boundary conditions on the deforming free surface. Two three-dimensional test problems are used to demonstrate the performance of the method: a liquid bridge problem and the formation of a fluid droplet.

A Distributed Co-Operative Problem Solving Environment

Scientific research is often multidisciplinary in nature and hence large projects are frequently collaborative with participants from several separate research centres. Rather than being restricted to infrequent dissemination of results and meetings a framework is described for embedding scientific computing applications within a collaborative problem solving environment. This allows users to combine their expertise in an interactive visual environment. Separate users can collaboratively steer and visualize data from a numerical simulation by embedding the simulation within the IRIS Explorer visualization system. By making use of this system the user has access to the COVISA toolkit which facilitates collaboration between separate users of IRIS Explorer. The flexibility of this system makes it straightforward to visualize and control as many aspects of the solution process as are desired.

Adaptive high-order finite element solution of transient elastohydrodynamic lubrication problems:

Abstract This article presents a new numerical method to solve transient line contact elastohydrodynamic lubrication (EHL) problems. A high-order discontinuous Galerkin (DG) finite element method is used for the spatial discretization, and the standard Crank-Nicolson method is employed to approximate the time derivative. An h-adaptivity method is used for grid adaptation with the time-stepping, and the penalty method is employed to handle the cavitation condition. The roughness model employed here is a simple indentation, which is located on the upper surface. Numerical results are presented comparing the DG method to standard finite difference (FD) techniques. It is shown that micro-EHL features are captured with far fewer degrees of freedom than when using low-order FD methods.

A model for virtual suturing in vascular surgery

Vascular surgery is a technically demanding surgical speciality, one component of which is the accurate placement of sutures through a diseased vessel wall. Minor errors can result in thrombosis and failure of the procedure. To develop the necessary skills takes many hours of practice, which, in the past, have been acquired at the operating table. Recreating a surgical environment using virtual tools presents a number of research challenges. Conventional collision detection methods fail for deformable bodies and do not provide a mechanism to scale the response over different regions at the same time. We have developed a threaded collision test allowing the mesh to be updated whilst guaranteeing a smooth force response for haptic devices. The finite element method (FEM) is adapted to allow multiple points of contact and the validity of this model is discussed with respect to known tissue behaviour. Following a preliminary examination by clinicians, a novel scheme is introduced for simulating more realistic force responses

Computational approaches for modelling elastohydrodynamic lubrication using multiphysics software.

Elastohydrodynamic lubrication modelling plays an important role in engineering design and analysis, since a number of important mechanical components operate under elastohydrodynamic lubrication conditions. In this article, methods are presented for solving both line and point contact cases using multiphysics software. The advantages, and the overheads, of using such an approach over developing highly specialised, bespoke software are highlighted. In order to calculate the deformation of the contacts three different methods are developed and their relative performance is assessed. The advantage of using a nested solution strategy has also been examined. The flexibility of the multiphysics software approach is highlighted in results involving a complex transient case modelling an involute gear.

A multilevel approach for obtaining locally optimal finite element meshes

In this paper we consider the adaptive finite element solution of a general class of variational problems using a combination of node insertion, node movement and edge swapping. The adaptive strategy that is proposed is based upon the construction of a hierarchy of locally optimal meshes starting with a coarse grid for which the location and connectivity of the nodes is optimized. This grid is then locally refined and the new mesh is optimized in the same manner. Results presented indicate that this approach is able to produce better meshes than those possible by more conventional adaptive strategies and in a relatively efficient manner.

A One-Field Monolithic Fictitious Domain Method for Fluid-Structure Interactions

Abstract In this article, we present a one-field monolithic fictitious domain (FD) method for simulation of general fluid–structure interactions (FSI). “One-field” means only one velocity field is solved in the whole domain, based upon the use of an appropriate L 2 projection. “Monolithic” means the fluid and solid equations are solved synchronously (rather than sequentially). We argue that the proposed method has the same generality and robustness as FD methods with distributed Lagrange multiplier (DLM) but is significantly more computationally efficient (because of one-field) whilst being very straightforward to implement. The method is described in detail, followed by the presentation of multiple computational examples in order to validate it across a wide range of fluid and solid parameters and interactions.

A quasi-cache-aware model for optimal domain partitioning in parallel geometric multigrid.

Stencil computations form the heart of numerical simulations to solve Partial Differential Equations using Finite Difference, Finite Element, and Finite Volume methods. Geometric Multigrid is an optimal O(N) , hierarchical tool employing stencil computations in its chief constituents, namely, smoothing, restriction, and interpolation. When Multigrid is parallelized over distributed‐shared memory architectures, traditionally, the domain partitioning creates cubic partitions of the mesh to minimize overall communication. Thus, the orthodox approach considers only load‐balancing and communication minimization for completely determining the domain partitioning. In this article, we show that these two factors are not sufficient to obtain optimal partitions for Parallel Geometric Multigrid. To this effect, we develop and validate a high level analytical model to show that “close to 2‐D” partitions for Geometric Multigrid can give higher performance than the partitions returned by the MPI_Dims_create() function which minimizes the communication volume by default. We quantify sub‐domain level cache‐misses in Parallel Geometric Multigrid and obtain families of optimal domain partitions. We conclude that the sub‐domain level cache‐misses for the application‐specific stencil computational kernel and communicated planes should be taken into account in addition to communication minimization/load‐balance to obtain optimal partitions for Parallel Geometric Multigrid.

A cache-aware approach to domain decomposition for stencil-based codes

Partial Differential Equations (PDEs) lie at the heart of numerous scientific simulations depicting physical phenomena. The parallelization of such simulations introduces additional performance penalties in the form of local and global synchronization among cooperating processes. Domain decomposition partitions the largest shareable data structures into sub-domains and attempts to achieve perfect load balance and minimal communication. Up to now research efforts to optimize spatial and temporal cache reuse for stencil-based PDE discretizations (e.g. finite difference and finite element) have considered sub-domain operations after the domain decomposition has been determined. We derive a cache-oblivious heuristic that minimizes cache misses at the sub-domain level through a quasi-cache-directed analysis to predict families of high performance domain decompositions in structured 3-D grids. To the best of our knowledge this is the first work to optimize domain decompositions by analyzing cache misses - thus connecting single core parameters (i.e. cache-misses) to true multicore parameters (i.e. domain decomposition). We analyze the trade-offs in decreasing cache-misses through such decompositions and increasing the dynamic bandwidth-per-core. The limitation of our work is that currently, it is applicable only to structured 3-D grids with cuts parallel to the Cartesian Axes. We emphasize and conclude that there is an imperative need to re-think domain decompositions in this constantly evolving multi-core era.