Application of a three-field nonlinear fluid-structure formulation to the prediction of the aeroelastic parameters of an F-16 fighter

We overview a three-field formulation of coupled fluid-structure interaction problems where the flow is modeled by the arbitrary Lagrangian-Eulerian form of either the Euler or Navier-Stokes equations, the structure is represented by a detailed finite element (FE) model, and the fluid grid is unstructured, dynamic, and constructed by a robust structure analogy method. We discuss the latest advances in the computational algorithms associated with this approach for modeling aeroelastic problems. We apply the three-field nonlinear computational framework to the prediction of the aeroelastic frequencies and damping coefficients of an F-16 configuration in various subsonic, transonic, and supersonic regimes. We consider for this purpose both the popular two-dimensional typical wing section model and a detailed three-dimensional FE model of the structure, and compare in both cases the obtained numerical results with flight test data. We comment on the advantages and shortfalls of both approaches, and on the feasibility as well as the merit of the three-field formulation of nonlinear aeroelasticity for the extraction of flutter envelopes.     

Coupling of a nonlinear finite element structural method with a Navier-Stokes solver

A new three-dimensional viscous aeroelastic solver is developed in the present work. A well validated full Navier-Stokes code is coupled with a nonlinear finite element plate model. Implicit coupling between the computational fluid dynamics and structural solvers is achieved using a subiteration approach. Computations of several benchmark static and dynamic plate problems are used to validate the finite element portion of the code. This coupled aeroelastic scheme is then applied to the problem of three-dimensional panel flutter. Inviscid and viscous supersonic results match previous computations using the same aerodynamic method coupled with a finite difference structural solver. For the case of subsonic flow, multiple solutions consisting of static, upward and downward deflections of the panel are discussed. The particular solution obtained is shown to be sensitive to the cavity pressure specified underneath the panel.     

Reliability-based shape optimization of structures undergoing fluid–structure interaction phenomena

Fluid–structure interaction phenomena are often roughly approximated when the stochastic nature of a system is considered in the design optimization process, leading to potentially significant epistemic uncertainty. In this paper, after reviewing the state-of-the-art methods in robust and reliability-based design optimization of problems undergoing fluid–structure interaction phenomena, a computational framework is presented that integrates a high-fidelity aeroelastic model into reliability-based design optimization. The design optimization problem is formulated pursuant to the reliability index and performance measure approaches. The system reliability is evaluated by a first-order reliability analysis method. The steady-state aeroelastic problem is described by a three-field formulation and solved by a staggered procedure, coupling a potentially detailed structural finite element model and a finite volume discretization of the Euler flow. The design and imperfection sensitivities are computed by evaluating the analytically derived direct and adjoint coupled aeroelastic sensitivity equations. The computational framework is verified by the optimization of three-dimensional wing structures. The lift-to-drag ratio is maximized, subject to stress constraints, by varying shape, thickness, and material properties. Uncertainties in structural parameters, including design parameters, operating conditions, and modeling uncertainties are considered. The results demonstrate the need for reliability-based optimization methods, for the design of structures undergoing fluid–structure interaction phenomena, and the applicability of the proposed framework to realistic design problems. Comparing the optimization results for different levels of uncertainty shows the importance of accounting for uncertainties in a quantitative manner.     

Conceptual design of aeroelastic structures by topology optimization

A topology optimization methodology is presented for the conceptual design of aeroelastic structures accounting for the fluid–structure interaction. The geometrical layout of the internal structure, such as the layout of stiffeners in a wing, is optimized by material topology optimization. The topology of the wet surface, that is, the fluid–structure interface, is not varied. The key components of the proposed methodology are a Sequential Augmented Lagrangian method for solving the resulting large-scale parameter optimization problem, a staggered procedure for computing the steady-state solution of the underlying nonlinear aeroelastic analysis problem, and an analytical adjoint method for evaluating the coupled aeroelastic sensitivities. The fluid–structure interaction problem is modeled by a three-field formulation that couples the structural displacements, the flow field, and the motion of the fluid mesh. The structural response is simulated by a three-dimensional finite element method, and the aerodynamic loads are predicted by a three-dimensional finite volume discretization of a nonlinear Euler flow. The proposed methodology is illustrated by the conceptual design of wing structures. The optimization results show the significant influence of the design dependency of the loads on the optimal layout of flexible structures when compared with results that assume a constant aerodynamic load.     

Strong discontinuities in partially saturated poroplastic solids

This paper considers the analysis and numerical simulation of strong discontinuities in partially saturated solids. The goal is to study observed localized failures in such media like shear bands and similar. The developments consider a fully coupled partially saturated elastoplastic model for the (continuum) bulk response of the solid formulated in effective stresses, identifying the necessary mathematical conditions for the appearance of strong discontinuities (that is, discontinuities in the displacement field leading to singular strains) as well as the proper treatment for the fields characterizing the flow of the different fluid phases, namely, the fluid contents of these phases and their individual pore pressures. The geometrically linear range of infinitesimal strains is considered. These developments allow the formulation of multiphase cohesive laws along the strong discontinuity, capturing in this way the coupled localized dissipation observed in the aforementioned failures. Furthermore, the paper also presents the formulation of enhanced finite elements capturing all these discontinuous solutions in general unstructured meshes. In particular, the finite elements capture the strong discontinuity through the proper enhancements of the discrete element strains, allowing for a complete local resolution of these effects. This results in a particularly efficient computational approach, easily accommodated in an existing finite element code. Different representative numerical simulations are presented illustrating the performance of the proposed formulation, as well as its use in practical applications like the modeling of the excavation of tunnels in variably saturated media.     

Efficient nonlinear static aeroelastic wing analysis

The objective of this work is to demonstrate a computationally efficient, high-fidelity, integrated static aeroelastic analysis procedure. The aerodynamic analysis consists of solving the nonlinear Euler equations by using an upwind cell-centered finite-volume scheme on unstructured tetrahedral meshes. The use of unstructured grids enhances the discretization of irregularly shaped domains and the interaction compatibility with the wing structure. The structural analysis utilizes finite elements to model the wing so that accurate structural deflections are obtained and allows the capability for computing detailed stress information for the configuration. Parameters are introduced to control the interaction of the computational fluid dynamics and structural analyses; these control parameters permit extremely efficient static aeroelastic computations. To demonstrate and evaluate this procedure, static aeroelastic analysis results for a flexible wing in low subsonic, high subsonic (subcritical), transonic (supercritical), and supersonic flow conditions are presented.     

Time-accurate Navier-Stokes simulation of vortex convection using an unstructured dynamic mesh procedure

A two-dimensional Navier-Stokes flow solver is developed for the simulation of unsteady flows on unstructured adaptive meshes. The solver is based on a second-order accurate implicit time integration using a point Gauss-Seidel relaxation scheme and a dual time-step subiteration. A vertex-centered, finite-volume discretization is used in conjunction with Roe’s flux-difference splitting. The Spalart-Allmaras one equation model is employed for the simulation of turbulence. An unsteady solution-adaptive dynamic mesh scheme is used by adding and deleting mesh points to take account of spatial and temporal variations of the flowfield. Unsteady viscous flow for a traveling vortex in a free stream is simulated to validate the accuracy of the dynamic mesh adaptation procedure. Flow around a circular cylinder and two blade-vortex interaction problems are investigated for demonstration of the present method. Computed results show good agreement with existing experimental and computational results. It was found that unsteady time-accurate viscous flows can be accurately simulated using the present unstructured dynamic mesh adaptation procedure.     

Reliability-based design optimization of aeroelastic structures

Aeroelastic phenomena are most often either ignored or roughly approximated when uncertainties are considered in the design optimization process of structures subject to aerodynamic loading, affecting the quality of the optimization results. Therefore, a design methodology is proposed that combines reliability-based design optimization and high-fidelity aeroelastic simulations for the analysis and design of aeroelastic structures. To account for uncertainties in design and operating conditions, a first-order reliability method (FORM) is employed to approximate the system reliability. To limit model uncertainties while accounting for the effects of given uncertainties, a high-fidelity nonlinear aeroelastic simulation method is used. The structure is modelled by a finite element method, and the aerodynamic loads are predicted by a finite volume discretization of a nonlinear Euler flow. The usefulness of the employed reliability analysis in both describing the effects of uncertainties on a particular design and as a design tool in the optimization process is illustrated. Though computationally more expensive than a deterministic optimum, due to the necessity of solving additional optimization problems for reliability analysis within each step of the broader design optimization procedure, a reliability-based optimum is shown to be an improved design. Conventional deterministic aeroelastic tailoring, which exploits the aeroelastic nature of the structure to enhance performance, is shown to often produce designs that are sensitive to variations in system or operational parameters.     

An Unstructured Finite Volume Approach for Structural Dynamics in Response to Fluid Motions

A new cell-vortex unstructured finite volume method for structural dynamics is assessed for simulations of structural dynamics in response to fluid motions. A robust implicit dual-time stepping method is employed to obtain time accurate solutions. The resulting system of algebraic equations is matrix-free and allows solid elements to include structure thickness, inertia, and structural stresses for accurate predictions of structural responses and stress distributions. The method is coupled with a fluid dynamics solver for fluid-structure interaction, providing a viable alternative to the finite element method for structural dynamics calculations. A mesh sensitivity test indicates that the finite volume method is at least of second-order accuracy. The method is validated by the problem of vortex-induced vibration of an elastic plate with different initial conditions and material properties. The results are in good agreement with existing numerical data and analytical solutions. The method is then applied to simulate a channel flow with an elastic wall. The effects of wall inertia and structural stresses on the fluid flow are investigated.     

Combined finite volume–finite element method for shallow water equations

This paper is concerned with solving the viscous and inviscid shallow water equations. The numerical method is based on second-order finite volume–finite element (FV–FE) discretization: the convective inviscid terms of the shallow water equations are computed by a finite volume method, while the diffusive viscous terms are computed with a finite element method. The method is implemented on unstructured meshes. The inviscid fluxes are evaluated with the approximate Riemann solver coupled with a second-order upwind reconstruction. Herein, the Roe and the Osher approximate Riemann solvers are used respectively and a comparison between them is made. Appropriate limiters are used to suppress spurious oscillations and the performance of three different limiters is assessed. Moreover, the second-order conforming piecewise linear finite elements are used. The second-order TVD Runge–Kutta method is applied to the time integration. Verification of the method for the full viscous system and the inviscid equations is carried out. By solving an advection–diffusion problem, the performance assessment for the FV–FE method, the full finite volume method, and the discontinuous Galerkin method is presented.     

Dispersive and Dissipative Properties of Discontinuous Galerkin Finite Element Methods for the Second-Order Wave Equation

Discontinuous Galerkin finite element methods (DGFEM) offer certain advantages over standard continuous finite element methods when applied to the spatial discretisation of the acoustic wave equation. For instance, the mass matrix has a block diagonal structure which, used in conjunction with an explicit time stepping scheme, gives an extremely economical scheme for time domain simulation. This feature is ubiquitous and extends to other time-dependent wave problems such as Maxwell’s equations. An important consideration in computational wave propagation is the dispersive and dissipative properties of the discretisation scheme in comparison with those of the original system. We investigate these properties for two popular DGFEM schemes: the interior penalty discontinuous Galerkin finite element method applied to the second-order wave equation and a more general family of schemes applied to the corresponding first order system. We show how the analysis of the multi-dimensional case may be reduced to consideration of one-dimensional problems. We derive the dispersion error for various schemes and conjecture on the generalisation to higher order approximation in space     

Simulation of a viscous compressible flow past a circular cylinder with high-order discontinuous Galerkin methods

This paper is devoted to the simulation of viscous compressible flows with high-order accurate discontinuous Galerkin methods on bidimensional unstructured meshes. The formulation for the solution of the Navier–Stokes equations is due to Oden et al. [An hp-adaptive discontinuous finite element method for computational fluid dynamics. PhD thesis, The University of Texas at Austin, 1997; J Comput Phys 1998;146:491–519]. It involves a weak imposition of continuity conditions on the state variables and on fluxes across interelement boundaries. It does not make use of any auxiliary variables and then the cost for the implementation is reasonable. The method is coupled with a limiting procedure recently developed by the authors to suppress oscillations near large gradients. The limiter is totally free of problem dependence and maintains the convergence order for errors measured in the L¹-norm. This paper presents detailed numerical results of a viscous compressible flow past a circular cylinder at a Reynolds number of 100 for the cases of subsonic and supersonic regimes. The proposed simulations suggest that the method is very robust and is able to produce very accurate results on unstructured meshes.     

An unfitted hp-adaptive finite element method based on hierarchical B-splines for interface problems of complex geometry

Generating finite element discretizations with direct interface parameterizations constitutes a considerable computational expense in case of complex interface geometries. The paper at hand introduces a B-spline finite element method, which circumvents parameterization of interfaces and offers fast and easy meshing irrespective of the geometric complexity involved. Its core idea is the adaptive approximation of discontinuities by hierarchical grid refinement, which adds several levels of local basis functions in the close vicinity of interfaces, but unfitted to their exact location, so that a simple regular grid of knot span elements can be maintained. Numerical experiments show that an hp-refinement strategy, which simultaneously increases the polynomial degree of the B-spline basis and the levels of refinement around interfaces, achieves exponential rates of convergence despite the presence of discontinuities. It is also demonstrated that the hierarchical B-spline FEM can be used to transfer the recently introduced Finite Cell concept to geometrically nonlinear problems. Its computational performance, imposition of unfitted boundary conditions and fast hierarchical grid generation are illustrated for a set of benchmark problems in one, two and three dimensions, and the advantages of the regular grid approach for complex geometries are demonstrated by the geometrically nonlinear simulation of a voxel based foam composite.     

Shape design optimization of stationary fluid-structure interaction problems with large displacements and turbulence

This paper presents general and efficient methods for analysis and gradient based shape optimization of systems characterized as strongly coupled stationary fluid-structure interaction (FSI) problems. The incompressible fluid flow can be laminar or turbulent and is described using the Reynolds-averaged Navier-Stokes equations (RANS) together with the algebraic Baldwin–Lomax turbulence model. The structure may exhibit large displacements due to the interaction with the fluid domain, resulting in geometrically nonlinear structural behaviour and nonlinear interface coupling conditions. The problem is discretized using Galerkin and Streamline-Upwind/Petrov–Galerkin finite element methods, and the resulting nonlinear equations are solved using Newtons method. Due to the large displacements of the structure, an efficient update algorithm for the fluid mesh must be applied, leading to the use of an approximate Jacobian matrix in the solution routine. Expressions for Design Sensitivity Analysis (DSA) are derived using the direct differentiation approach, and the use of an inexact Jacobian matrix in the analysis leads to an iterative but very efficient scheme for DSA. The potential of gradient based shape optimization of fluid flow and FSI problems is illustrated by several examples.     

Three dimensional discontinuous Galerkin methods for Euler equations on adaptive conforming meshes

In the numerical simulation of three dimensional fluid dynamical equations, the huge computational quantity is a main challenge. In this paper, the discontinuous Galerkin (DG) finite element method combined with the adaptive mesh refinement (AMR) is studied to solve the three dimensional Euler equations based on conforming unstructured tetrahedron meshes, that is according the equation solution variation to refine and coarsen grids so as to decrease total mesh number. The four space adaptive strategies are given and analyzed their advantages and disadvantages. The numerical examples show the validity of our methods.     

Multigrid for finite element discretizations of aerodynamics equations

A multigrid algorithm for the solution of a finite element stabilized discretization of compressible fluid dynamics equations on unstructured grids is described. The solution of the stationary problems is sought by time-stepping and a linearization of the nonlinear discrete systems leads to a very large system of linear equations. These systems are ill-conditioned and require efficient computational procedures. The numerical experiments for Navier-Stokes and Euler systems are presented. The method can be easily included in a parallel library as a preconditioner.     

Comparative study of different numerical approaches in space–time CESE framework for high-fidelity flow simulations

With the advancement of computer hardware, the trend of research in computational fluid dynamics is moving towards development of highly accurate, unstructured-mesh compatible, robust and efficient numerical methods for simulating problems involving strong transient effects and relatively complex geometries as well as physics. The space–time conservation element and solution element method is a genuinely multi-dimensional, unstructured-mesh compatible numerical framework, which was built from a consistent and synergetic integration of conservation laws in the space–time domain to avoid the limitations of conventional schemes, such as the use of 1-D flux reconstruction with a Riemann solver. It has been shown that the framework can be used for time-accurate simulations of a variety of problems involving unsteady waves, strong flow discontinuities, and their interactions with remarkable accuracy. However, this method at its current state has encountered the challenge in balancing the robustness and numerical accuracy when highly stretched meshes were used in viscous flow simulation. In this paper, we briefly discuss various numerical approaches developed for this framework thus far as well as their strengths and weaknesses, and conduct a comparative study of their numerical accuracies using some 2-D viscous benchmark test cases. The application of this method in realistic, complex 3-D problems is also included here to demonstrate its computational efficiency in large-scale computing.     

Numerical approximation of flow induced vibrations of channel walls

In this paper the numerical method for solution of an aeroelastic model describing the interactions of air flow with vocal folds is described. The flow is modelled by the incompressible Navier–Stokes equations spatially discretized with the aid of the stabilized finite element method. The motion of the computational domain is treated with the aid of Arbitrary Lagrangian–Eulerian method. The structure motion is described by an equivalent system with two degrees of freedom governed by a system of ordinary differential equations and discretized in time with the aid of an implicit multistep method and strongly coupled with the flow model. The influence of inlet/outlet boundary conditions is studied. The numerical analysis is performed and compared to the related results from literature.     

Parallel adaptive simulations of dynamic fracture events

Finite element simulations of dynamic fracture problems usually require very fine discretizations in the vicinity of the propagating stress waves and advancing crack fronts, while coarser meshes can be used in the remainder of the domain. This need for a constantly evolving discretization poses several challenges, especially when the simulation is performed on a parallel computing platform. To address this issue, we present a parallel computational framework developed specifically for unstructured meshes. This framework allows dynamic adaptive refinement and coarsening of finite element meshes and also performs load balancing between processors. We demonstrate the capability of this framework, called ParFUM, using two-dimensional structural dynamic problems involving the propagation of elastodynamic waves and the spontaneous initiation and propagation of cracks through a domain discretized with triangular finite elements.     

基于动力学仿真的系留气球鼻锥有限元分析

针对系留气球进行了动力学仿真分析,在此基础上对一种用于固定球体的新型鼻锥结构进行了结构有限元分析,以确定其强度与刚度.首先采用计算流体力学CFD求得特定风速下系留气球所受的气动力,随后通过多体动力学软件Adams进行动力学仿真分析,确定作用在鼻锥上载荷的大小,并以此作为有限元分析的载荷边界条件;然后采用有限元分析的方法对鼻锥结构进行静力学和动力学分析;最后确定极限风速下艇首与鼻锥连接处的变形、载荷及应力情况.通过分析,为新型鼻锥结构进一步的设计改进与优化提供了参考依据.