Dynamic fluid–structure interaction using finite volume unstructured mesh procedures

A three-dimensional finite volume, unstructured mesh (FV-UM) method for dynamic fluid–structure interaction (DFSI) is described. Fluid structure interaction, as applied to flexible structures, has wide application in diverse areas such as flutter in aircraft, wind response of buildings, flows in elastic pipes and blood vessels. It involves the coupling of fluid flow and structural mechanics, two fields that are conventionally modelled using two dissimilar methods, thus a single comprehensive computational model of both phenomena is a considerable challenge. Until recently work in this area focused on one phenomenon and represented the behaviour of the other more simply. More recently, strategies for solving the full coupling between the fluid and solid mechanics behaviour have been developed. A key contribution has been made by Farhat et al. [Int. J. Numer. Meth. Fluids 21 (1995) 807] employing FV-UM methods for solving the Euler flow equations and a conventional finite element method for the elastic solid mechanics and the spring based mesh procedure of Batina [AIAA paper 0115, 1989] for mesh movement. In this paper, we describe an approach which broadly exploits the three field strategy described by Farhat for fluid flow, structural dynamics and mesh movement but, in the context of DFSI, contains a number of novel features: • a single mesh covering the entire domain, • a Navier–Stokes flow, • a single FV-UM discretisation approach for both the flow and solid mechanics procedures, • an implicit predictor–corrector version of the Newmark algorithm, • a single code embedding the whole strategy.

Dynamic fluid-structure interactions using finite volume unstructured mesh procedures

A three-dimensional finite volume, unstructured mesh (FV-UM) method for dynamic fluid–structure interaction (DFSI) is described. Fluid structure interaction, as applied to flexible structures, has wide application in diverse areas such as flutter in aircraft, wind response of buildings, flows in elastic pipes and blood vessels. It involves the coupling of fluid flow and structural mechanics, two fields that are conventionally modelled using two dissimilar methods, thus a single comprehensive computational model of both phenomena is a considerable challenge. Until recently work in this area focused on one phenomenon and represented the behaviour of the other more simply. More recently, strategies for solving the full coupling between the fluid and solid mechanics behaviour have been developed. A key contribution has been made by Farhat et al. [Int. J. Numer. Meth. Fluids 21 (1995) 807] employing FV-UM methods for solving the Euler flow equations and a conventional finite element method for the elastic solid mechanics and the spring based mesh procedure of Batina [AIAA paper 0115, 1989] for mesh movement. In this paper, we describe an approach which broadly exploits the three field strategy described by Farhat for fluid flow, structural dynamics and mesh movement but, in the context of DFSI, contains a number of novel features: a single mesh covering the entire domain, a Navier–Stokes flow, a single FV-UM discretisation approach for both the flow and solid mechanics procedures, an implicit predictor–corrector version of the Newmark algorithm, a single code embedding the whole strategy.

A two dimensional unstructured staggered mesh method with special treatment of tangential velocity

A two dimensional staggered unstructured discretisation scheme for the solution of fluid flow problems has been developed. This scheme stores and solves the velocity vector resolutes normal and parallel to each cell face and other scalar variables (pressure, temperature) are stored at cell centres. The coupled momentum; continuity and energy equations are solved, using the well known pressure correction algorithm SIMPLE. The method is tested for accuracy and convergence behaviour against standard cell-centre solutions in a number of benchmark problems: The Lid-Driven Cavity, Natural Convection in a Cavity and the Melting of Gallium in a rectangular domain.

Numerical simulation of incompressible flow problems using an unstructured staggered mesh method

A number of two dimensional staggered unstructured discretisation schemes for the solution of fluid flow and heat transfer problems have been developed. All schemes store and solve velocity vector components at cell faces with scalar variables solved at cell centres. The velocity is resolved into face-normal and face-parallel components and the various schemes investigated differ in the treatment of the parallel component. Steady-state and time-dependent fluid flow and thermal energy equations are solved with the well known pressure correction scheme, SIMPLE, employed to couple continuity and momentum. The numerical methods developed are tested on well known benchmark cases: the Lid-Driven Cavity, Natural Convection in a Cavity and Melting of Gallium in a rectangular domain. The results obtained are shown to be comparable to benchmark, but with accuracy dependent on scheme selection.

A high-order arbitrarily unstructured finite-volume model of the global atmosphere: tests solving the shallow-water equations

Simulations of the global atmosphere for weather and climate forecasting require fast and accurate solutions and so operational models use high-order finite differences on regular structured grids. This precludes the use of local refinement; techniques allowing local refinement are either expensive (eg. high-order finite element techniques) or have reduced accuracy at changes in resolution (eg. unstructured finite-volume with linear differencing). We present solutions of the shallow-water equations for westerly flow over a mid-latitude mountain from a finite-volume model written using OpenFOAM. A second/third-order accurate differencing scheme is applied on arbitrarily unstructured meshes made up of various shapes and refinement patterns. The results are as accurate as equivalent resolution spectral methods. Using lower order differencing reduces accuracy at a refinement pattern which allows errors from refinement of the mountain to accumulate and reduces the global accuracy over a 15 day simulation. We have therefore introduced a scheme which fits a 2D cubic polynomial approximately on a stencil around each cell. Using this scheme means that refinement of the mountain improves the accuracy after a 15 day simulation. This is a more severe test of local mesh refinement for global simulations than has been presented but a realistic test if these techniques are to be used operationally. These efficient, high-order schemes may make it possible for local mesh refinement to be used by weather and climate forecast models.

A cell by cell anisotropic adaptive mesh ALE method

A cell by cell anisotropic adaptive mesh Arbitrary Lagrangian Eulerian (ALE) method for the solution of the Euler equations is described. An efficient approach to equipotential mesh relaxation on anisotropically refined meshes is developed. Results for two test problems are presented.

Predicting mesh density for adaptive modelling of the global atmosphere

The shallow water equations are solved using a mesh of polygons on the sphere, which adapts infrequently to the predicted future solution. Infrequent mesh adaptation reduces the cost of adaptation and load-balancing and will thus allow for more accurate mapping on adaptation. We simulate the growth of a barotropically unstable jet adapting the mesh every 12 h. Using an adaptation criterion based largely on the gradient of the vorticity leads to a mesh with around 20 per cent of the cells of a uniform mesh that gives equivalent results. This is a similar proportion to previous studies of the same test case with mesh adaptation every 1–20 min. The prediction of the mesh density involves solving the shallow water equations on a coarse mesh in advance of the locally refined mesh in order to estimate where features requiring higher resolution will grow, decay or move to. The adaptation criterion consists of two parts: that resolved on the coarse mesh, and that which is not resolved and so is passively advected on the coarse mesh. This combination leads to a balance between resolving features controlled by the large-scale dynamics and maintaining fine-scale features.

Scale-invariant moving finite elements for nonlinear partial differential equations in two dimensions

A scale-invariant moving finite element method is proposed for the adaptive solution of nonlinear partial differential equations. The mesh movement is based on a finite element discretisation of a scale-invariant conservation principle incorporating a monitor function, while the time discretisation of the resulting system of ordinary differential equations is carried out using a scale-invariant time-stepping which yields uniform local accuracy in time. The accuracy and reliability of the algorithm are successfully tested against exact self-similar solutions where available, and otherwise against a state-of-the-art h-refinement scheme for solutions of a two-dimensional porous medium equation problem with a moving boundary. The monitor functions used are the dependent variable and a monitor related to the surface area of the solution manifold. (c) 2005 IMACS. Published by Elsevier B.V. All rights reserved.

An efficient shock capturing scheme for the pseudo-unsteady equations representing steady inviscid flow

An efficient algorithm is presented for the solution of the steady Euler equations of gas dynamics. The scheme is based on solving linearised Riemann problems approximately and in more than one dimension incorporates operator splitting. The scheme is applied to a standard test problem of flow down a channel containing a circular arc bump for three different mesh sizes.

A non-uniform mesh scheme for compressible flow

A non-uniform mesh scheme is presented for the computation of compressible flows governed by the Euler equations of gas dynamics. The scheme is based on flux-difference splitting and represents an extension of a similar scheme designed for uniform meshes. The numerical results demonstrate that little, if any, spurious oscillation occurs as a result of the non-uniformity of the mesh; and importantly, shock speeds are computed correctly.

Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations

In this paper, we extend to the time-harmonic Maxwell equations the p-version analysis technique developed in [R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version, SIAM J. Numer. Anal., 49 (2011), 264-284] for Trefftz-discontinuous Galerkin approximations of the Helmholtz problem. While error estimates in a mesh-skeleton norm are derived parallel to the Helmholtz case, the derivation of estimates in a mesh-independent norm requires new twists in the duality argument. The particular case where the local Trefftz approximation spaces are built of vector-valued plane wave functions is considered, and convergence rates are derived.

Numerical conformal mapping and mesh generation for polygonal and multiply-connected regions

Details are given of a boundary-fitted mesh generation method for use in modelling free surface flow and water quality. A numerical method has been developed for generating conformal meshes for curvilinear polygonal and multiply-connected regions. The method is based on the Cauchy-Riemann conditions for the analytic function and is able to map a curvilinear polygonal region directly onto a regular polygonal region, with horizontal and vertical sides. A set of equations have been derived for determining the lengths of these sides and the least-squares method has been used in solving the equations. Several numerical examples are presented to illustrate the method.

A depth-integrated 2D coastal and estuarine model with conformal boundary-fitted mesh generation

Details are given of the development and application of a 2D depth-integrated, conformal boundary-fitted, curvilinear model for predicting the depth-mean velocity field and the spatial concentration distribution in estuarine and coastal waters. A numerical method for conformal mesh generation, based on a boundary integral equation formulation, has been developed. By this method a general polygonal region with curved edges can be mapped onto a regular polygonal region with the same number of horizontal and vertical straight edges and a multiply connected region can be mapped onto a regular region with the same connectivity. A stretching transformation on the conformally generated mesh has also been used to provide greater detail where it is needed close to the coast, with larger mesh sizes further offshore, thereby minimizing the computing effort whilst maximizing accuracy. The curvilinear hydrodynamic and solute model has been developed based on a robust rectilinear model. The hydrodynamic equations are approximated using the ADI finite difference scheme with a staggered grid and the solute transport equation is approximated using a modified QUICK scheme. Three numerical examples have been chosen to test the curvilinear model, with an emphasis placed on complex practical applications

A moving mesh approach for modelling avascular tumour growth

A key step in many numerical schemes for time-dependent partial differential equations with moving boundaries is to rescale the problem to a fixed numerical mesh. An alternative approach is to use a moving mesh that can be adapted to focus on specific features of the model. In this paper we present and discuss two different velocity-based moving mesh methods applied to a two-phase model of avascular tumour growth formulated by Breward et al. (2002) J. Math. Biol. 45(2), 125-152. Each method has one moving node which tracks the moving boundary. The first moving mesh method uses a mesh velocity proportional to the boundary velocity. The second moving mesh method uses local conservation of volume fraction of cells (masses). Our results demonstrate that these moving mesh methods produce accurate results, offering higher resolution where desired whilst preserving the balance of fluxes and sources in the governing equations.

Mesh adaptation on the sphere using optimal transport and the numerical solution of a Monge-Ampère type equation

An equation of Monge-Ampère type has, for the first time, been solved numerically on the surface of the sphere in order to generate optimally transported (OT) meshes, equidistributed with respect to a monitor function. Optimal transport generates meshes that keep the same connectivity as the original mesh, making them suitable for r-adaptive simulations, in which the equations of motion can be solved in a moving frame of reference in order to avoid mapping the solution between old and new meshes and to avoid load balancing problems on parallel computers. The semi-implicit solution of the Monge-Ampère type equation involves a new linearisation of the Hessian term, and exponential maps are used to map from old to new meshes on the sphere. The determinant of the Hessian is evaluated as the change in volume between old and new mesh cells, rather than using numerical approximations to the gradients. OT meshes are generated to compare with centroidal Voronoi tesselations on the sphere and are found to have advantages and disadvantages; OT equidistribution is more accurate, the number of iterations to convergence is independent of the mesh size, face skewness is reduced and the connectivity does not change. However anisotropy is higher and the OT meshes are non-orthogonal. It is shown that optimal transport on the sphere leads to meshes that do not tangle. However, tangling can be introduced by numerical errors in calculating the gradient of the mesh potential. Methods for alleviating this problem are explored. Finally, OT meshes are generated using observed precipitation as a monitor function, in order to demonstrate the potential power of the technique.