Charbel Farhat

Load and motion transfer algorithms for fluid/structure interaction problems with non-matching discrete interfaces: Momentum and energy conservation, optimal discretization and application to aeroelasticity

The prediction of many fluid/structure interaction phenomena requires solving simultaneously the coupled fluid and structural equations of equilibrium with an appropriate set of interface boundary conditions. In this paper, we consider the realistic situation where the fluid and structure subproblems have different resolution requirements and their computational domains have non-matching discrete interfaces, and address the proper discretization of the governing interface boundary conditions. We present and overview new and common algorithms for converting the fluid pressure and stress fields at the fluid/structure interface into a structural load, and for transferring the structural motion to the fluid system. We discuss the merits of these algorithms in terms of conservation properties and solution accuracy, and distinguish between theoretically important and practically significant issues. We validate our claims and illustrate our conclusions with several transient aeroelastic simulations.

Partitioned analysis of coupled mechanical systems

Abstract This is a tutorial article that reviews the use of partitioned analysis procedures for the analysis of coupled dynamical systems. Attention is focused on the computational simulation of systems in which a structure is a major component. Important applications in that class are provided by thermomechanics, fluid–structure interaction and control–structure interaction. In the partitioned solution approach, systems are spatially decomposed into partitions. This decomposition is driven by physical or computational considerations. The solution is separately advanced in time over each partition. Interaction effects are accounted for by transmission and synchronization of coupled state variables. Recent developments in the use of this approach for multilevel decomposition aimed at massively parallel computation are discussed.

Torsional springs for two-dimensional dynamic unstructured fluid meshes

Abstract Dynamic fluid grids are commonly used for the solution of flow problems with moving boundaries. They are often represented by a network of fictitious lineal springs that can become unreliable when the fluid mesh undergoes large displacements and/or deformations. In this paper, we propose to control the arbitrary motion of two-dimensional dynamic unstructured fluid grids with additional torsional springs. We show that such springs can be designed to prohibit the interpenetration of neighboring triangles, and therefore to provide the method of spring analogy with the robustness needed for enlarging its range of applications. We illustrate our new dynamic mesh motion algorithm with several examples that highlight its advantages in terms of robustness, quality, and performance.

Two efficient staggered algorithms for the serial and parallel solution of three-dimensional nonlinear transient aeroelastic problems

Abstract Partitioned procedures and staggered algorithms are often adopted for the solution of coupled fluid/structure interaction problems in the time domain. In this paper, we overview two sequential and parallel partitioned procedures that are popular in computational nonlinear aeroelasticity, and address their limitation in terms of accuracy and numerical stability. We propose two alternative serial and parallel staggered algorithms for the solution of coupled transient aeroelastic problems, and demonstrate their superior accuracy and computational efficiency with the flutter analysis of the AGARD Wing 445.6. We contrast our results with those computed by other investigators and validate them with experimental data.

The Discontinuous Enrichment Method

We propose a finite element based discretization method in which the standard polynomial field is enriched within each element by a nonconforming field that is added to it. The enrichment contains free-space solutions of the homogeneous differential equation that are not represented by the underlying polynomial field. Continuity of the enrichment across element interfaces is enforced weakly by Lagrange multipliers. In this manner, we expect to attain high coarse-mesh accuracy without significant degradation of conditioning, assuring good performance of the computation at any mesh resolution. Examples of application to acoustics and advection-diffusion are presented.

Optimal convergence properties of the FETI domain decomposition method

Abstract The Finite Element Tearing and Interconnecting (FETI) method is a practical and efficient domain decomposition (DD) algorithm for the solution of self-adjoint elliptic partial differential equations. For large-scale structural problems discretized with shell and beam elements, this method was found to outperform popular iterative algorithms and direct solvers on both serial and parallel computers, and to compare favorably with leading DD methods. In this paper, we discuss some numerical properties of the FETI method that were not addressed before. In particular, we show that the mathematical treatment of the floating subdomains and the specific conjugate projected gradient algorithm that characterize the FETI method are equivalent to the construction and solution of a coarse problem that propagates the error globally, accelerates convergence, and ensures a performance that is independent of the number of subdomains. We also show that when the interface problem is optimally preconditioned and the mesh is partitioned into well structured subdomains with good aspect ratios, the performance of the FETI method is also independent of the mesh size. However, we also argue that the FETI and other leading DD methods for unstructured problems lose in practice these scalability properties when the mesh contains junctures with rotational degrees of freedom, or the decomposition is irregular and characterized by arbitrary subdomain aspect ratios. Finally, we report that for realistic problems, optimal preconditioners are not necessarily computationally efficient and can be outperformed by non-optimal ones.

Interpolation Method for Adapting Reduced-Order Models and Application to Aeroelasticity

Reduced-order models are usually thought of as computationally inexpensive mathematical representations that offer the potential for near real-time analysis. Although most reduced-order models can operate in near real-time, their construction can be computationally expensive, as it requires accumulating a large number of system responses to input excitations. Furthermore, reduced-order models usually lack robustness with respect to parameter changes and therefore must often be rebuilt for each parameter variation. Together, these two issues underline the need for a fast and robust method for adapting precomputed reduced-order models to new sets of physical or modeling parameters. To this effect, this paper presents an interpolation method based on the Grassmann manifold and its tangent space at a point that is applicable to structural, aerodynamic, aeroelastic, and many other reduced-order models based on projection schemes. This method is illustrated here with the adaptation of computational-fluid-dynamics-based aeroelastic reduced-order models of complete fighter configurations to new values of the freestream Mach number. Good correlations with results obtained from direct reduced-order model reconstruction, high-fidelity nonlinear and linear simulations are reported, thereby highlighting the potential of the proposed reduced-order model adaptation method for near real-time aeroelastic predictions using precomputed reduced-order model databases.

Geometric conservation laws for flow problems with moving boundaries and deformable meshes, and their impact on aeroelastic computations

Abstract Numerical simulations of flow problems with moving boundaries commonly require the solution of the fluid equations on unstructured and deformable dynamic meshes. In this paper, we present a unified theory for deriving Geometric Conservation Laws (GCLs) for such problems. We consider several popular discretization methods for the spatial approximation of the flow equations including the Arbitrary Lagrangian-Eulerian (ALE) finite volume and finite element schemes, and space-time stabilized finite element formulations. We show that, except for the case of the space-time discretization method, the GCLs impose important constraints on the algorithms employed for time-integrating the semi-discrete equations governing the fluid and dynamic mesh motions. We address the impact of these constraints on the solution of coupled aeroelastic problems, and highlight the importance of the GCLs with an illustration of their effect on the computation of the transient aeroelastic response of a flat panel in transonic flow.

A simple and efficient automatic fem domain decomposer

Abstract The various forms of parallel numerical algorithms that speed up finite element computations are as different as the number of researchers working on the problem. However, most of the recently proposed concurrent computational strategies seem to have a common starting point, namely, domain decomposition. In this paper, a simple and efficient non-numerical algorithm for the automatic decomposition of an arbitrary finite element domain into a specified number of balanced subdomains is presented. It is shown that both the algorithm and its implementation are suitable for shared memory as well as local memory multiprocessors. Fortran 77 subroutines for the decomposer are given. They have proven to be extremely effective for the implementation of promising concurrent solution strategies on high performance architectures. It is hoped that the decomposer will relieve the burden of the preprocessing phase from the methods developers, so that they can concentrate on the numerical and synchronization issues of parallel computing.

Partitioned procedures for the transient solution of coupled aroelastic problems Part I: Model problem, theory and two-dimensional application

Abstract In order to predict the dynamic response of a flexible structure in a fluid flow, the equations of motion of the structure and the fluid must be solved simultaneously. In this paper we present several partitioned procedures for time-integrating this focus coupled problem and discuss their merits in terms of accuracy, stability, heterogeneous computing, I/O transfers, subcycling and parallel processing. All theoretical results are derived for a one-dimensional piston model problem with a compressible flow, because the complete three-dimensional aeroelastic problem is difficult to analyze mathematically. However, the insight gained from the analysis of the coupled piston problem and the conclusions drawn from its numerical investigation are confirmed with the numerical simulation of the two-dimensional transient aeroelastic response of a flexible panel in a transonic non-linear Euler flow regime.