On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far‐field boundary treatment

A reduced order model (ROM) based on the proper orthogonal decomposition (POD)/Galerkin projection method is proposed as an alternative discretization of the linearized compressible Euler equations. It is shown that the numerical stability of the ROM is intimately tied to the choice of inner product used to define the Galerkin projection. For the linearized compressible Euler equations, a symmetry transformation motivates the construction of a weighted L² inner product that guarantees certain stability bounds satisfied by the ROM. Sufficient conditions for well‐posedness and stability of the present Galerkin projection method applied to a general linear hyperbolic initial boundary value problem (IBVP) are stated and proven. Well‐posed and stable far‐field and solid wall boundary conditions are formulated for the linearized compressible Euler ROM using these more general results. A convergence analysis employing a stable penalty‐like formulation of the boundary conditions reveals that the ROM solution converges to the exact solution with refinement of both the numerical solution used to generate the ROM and of the POD basis. An a priori error estimate for the computed ROM solution is derived, and examined using a numerical test case. Published in 2010 by John Wiley & Sons, Ltd.     

Specially orthotropic panel flutter control using PID controller

Controlling the flutter speed of a specially orthotropic plate of rectangular cross-section through which an inviscid compressible fluid flows is considered. A state-space model of the coupled aeroelastic system helps us to determine flutter boundaries and provides a method for applying modern control theory to the problem. So in the present study, flow is modelled by piston theory and the coupled state-space model of panel motion and flow is solved. After that, the coupled system is controlled with Proportional-Integral-Derivative controller. This makes it possible to compute the flutter dynamic pressure and so the flutter velocity at which unstable plate oscillations occur as a function of the gain coefficient. It is found that by changing the gain coefficient the flutter of the plate can indeed be pushed to higher velocities.     

Robin‐Neumann transmission conditions for fluid‐structure coupling: Embedded boundary implementation and parameter analysis

Partitioned procedures are appealing for solving complex fluid‐structure interaction (FSI) problems, as they allow existing computational fluid dynamics (CFD) and computational structural dynamics algorithms and solvers to be combined and reused. However, for problems involving incompressible flow and strong added‐mass effect (eg, heavy fluid and slender structure), partitioned procedures suffer from numerical instability, which typically requires additional subiterations between the fluid and structural solvers, hence significantly increasing the computational cost. This paper investigates the use of Robin‐Neumann transmission conditions to mitigate the above instability issue. Firstly, an embedded Robin boundary method is presented in the context of projection‐based incompressible CFD and finite element–based computational structural dynamics. The method utilizes operator splitting and a modified ghost fluid method to enforce the Robin transmission condition on fluid‐structure interfaces embedded in structured non–body‐conforming CFD grids. The method is demonstrated and verified using the Turek and Hron benchmark problem, which involves a slender beam undergoing large transient deformation in an unsteady vortex‐dominated channel flow. Secondly, this paper investigates the effect of the combination parameter in the Robin transmission condition, ie, α_f, on numerical stability and solution accuracy. This paper presents a numerical study using the Turek and Hron benchmark problem and an analytical study using a simplified FSI model featuring an Euler‐Bernoulli beam interacting with a two‐dimensional incompressible inviscid flow. Both studies reveal a trade‐off between stability and accuracy: smaller values of α_f tend to improve numerical stability, yet deteriorate the accuracy of the partitioned solution. Using the simplified FSI model, the critical value of α_f that optimizes this trade‐off is derived and discussed.     

A rapid simulation of nano‐particle transport in a two‐dimensional human airway using POD/Galerkin reduced‐order models

An approach is proposed for the rapid prediction of nano‐particle transport and deposition in the human airway, which requires the solution of both the Navier–Stokes and advection–diffusion equations and for which computational efficiency is a challenge. The proposed method builds low‐order models that are representative of the fully coupled equations by means of the Galerkin projection and proper orthogonal decomposition technique. The obtained reduced‐order models (ROMs) are a set of ordinary differential equations for the temporal coefficients of the basis functions. The numerical results indicate that the ROMs are highly efficient for the computation (the speedup factor is approximately 3 × 10³) and have reasonable accuracy compared with the full model (relative error of ≈7 × 10^?3). Using ROMs, the deposition of particles is studied for 1≤d_n≤100 nm, where d_n is the diameter of a nano‐particle. The effectiveness of this approach is promising for applications of health risk assessment. Copyright © 2015 John Wiley & Sons, Ltd.     

A hybrid stabilization approach for reduced-order models of compressible flows with shock-vortex interaction

Model order reduction approaches, such as proper orthogonal decomposition (POD)-Galerkin projection, provide a systematic manner to construct Reduced-Order Models (ROM) from pregenerated high-fidelity datasets. The current study focuses on the stabilization of ROMs built from high-fidelity simulation data of a supersonic flow passing a circular cylinder, which features strong interactions between shockwaves and vortices. As shown in previous literatures and the current study, an implicit subspace correction (ISC) method is efficient in the stabilization of similar problems, but its accuracy is not consistent when applied on different ROMs; on the other hand, an eigenvalue reassignment (ER) method delivers superb accuracy when the mode number is small, but becomes too expensive and less robust as the number increases. A Hybrid method is proposed here to balance the computational cost while improving the overall robustness/accuracy in ROM stabilization. The Hybrid method first handles the majority of the modes using the ISC method and then applies the ER method to fine tune a smaller number of modes under a constraint for accuracy. Furthermore, when the typical L2 inner product is changed to a symmetry inner product in both POD computation and Galerkin projection, the performance of the stabilized ROMs is substantially improved for all methods.     

A non-linearly stable implicit finite element algorithm for hypersonic aerodynamics

A generalized curvilinear co-ordinate Taylor weak statement implicit finite element algorithm is developed for the two-dimensional and axisymmetric compressible Navier-Stokes equations for ideal and reacting gases. For accurate hypersonic simulation, air is modelled as a mixture of five perfect gases, i.e. molecular and atomic oxygen and nirogen as well as nitric oxide. The associated pressure is then determined via Newton solution of the classical chemical equilibrium equation system. The directional semi-discretization is achieved using an optimal metric data Galerkin finite element weak statement, on a developed ‘companion conservation law system’, permitting classical test and trial space definitions. Utilizing an implicit Runge-Kutta scheme, the terminal algorithm is then non-linearly stable, and second-order accurate in space and time on arbitrary curvilinear co-ordinates. Subsequently, a matrix tensor product factorization procedure permits an efficient numerical linear algebra handling for large Courant numbers. For ideal- and real-gas hypersonic flows, the algorithm generates essentially non-oscillatory numerical solutions in the presence of strong detached shocks and boundary layer inviscid flow interactions.     

A method for interpolating on manifolds structural dynamics reduced‐order models

A rigorous method for interpolating a set of parameterized linear structural dynamics reduced‐order models (ROMs) is presented. By design, this method does not operate on the underlying set of parameterized full‐order models. Hence, it is amenable to an online real‐time implementation. It is based on mapping appropriately the ROM data onto a tangent space to the manifold of symmetric positive‐definite matrices, interpolating the mapped data in this space and mapping back the result to the aforementioned manifold. Algorithms for computing the forward and backward mappings are offered for the case where the ROMs are derived from a general Galerkin projection method and the case where they are constructed from modal reduction. The proposed interpolation method is illustrated with applications ranging from the fast dynamic characterization of a parameterized structural model to the fast evaluation of its response to a given input. In all cases, good accuracy is demonstrated at real‐time processing speeds. Copyright © 2009 John Wiley & Sons, Ltd.     

On the a priori and a posteriori error analysis of a two‐fold saddle‐point approach for nonlinear incompressible elasticity

In this paper, we reconsider the a priori and a posteriori error analysis of a new mixed finite element method for nonlinear incompressible elasticity with mixed boundary conditions. The approach, being based only on the fact that the resulting variational formulation becomes a two‐fold saddle‐point operator equation, simplifies the analysis and improves the results provided recently in a previous work. Thus, a well‐known generalization of the classical Babu?ka–Brezzi theory is applied to show the well‐posedness of the continuous and discrete formulations, and to derive the corresponding a priori error estimate. In particular, enriched PEERS subspaces are required for the solvability and stability of the associated Galerkin scheme. In addition, we use the Ritz projection operator to obtain a new reliable and quasi‐efficient a posteriori error estimate. Finally, several numerical results illustrating the good performance of the associated adaptive algorithm are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

Vibration analysis of fluid–solid systems using a finite element displacement formulation

This report presents a finite element solution for the vibration interaction between an inviscid fluid and a solid. The equation of motion governing the inviscid fluid is expressed in terms of the displacements. This ensures that compatibility and equilibrium will be satisfied automatically along the interface of the coupled systems. To suppress circulation modes with non-zero energy, reduced integration is used when computing the element stiffness matrix contributed by the fluid. In addition, a projection is used on the element mass matrix in order to remove the spurious modes which result from the use of reduced integration. Numerical examples for both fluid and coupled fluid–solid systems are performed and the results are shown.     

A novel model for vibrations of nanotubes conveying nanoflow

In this paper, we have presented an innovative model for coupled vibrations of nanotubes conveying fluid by considering the small-size effects on the flow field. By this model, we have demonstrated that ignoring the small-size effects on flow field in a nano-scale fluid-structure interaction (FSI) problem may generate erroneous results. The nanotube has been modeled by Euler-Bernoulli plug-flow beam kinematic theory, and we have formulated the small-size effects on bulk viscosity and slip boundary conditions of nanoflow through Knudsen number (Kn), as a discriminant parameter. The divergence instability phenomenon has been observed, incorporating various flow regimes for liquids and gases. We have observed that including the effect of nanoflow viscosity, is not so influential on vibration of nanotubes conveying fluid, as compared with the results of vibration of nanotubes conveying an inviscid fluid; however, incorporating the nanoflow slip-boundary conditions hypothesis changes the results drastically, as compared to continuum flow models.     

Investigation of fluid–structure interactions using a velocity‐linked P2/P1 finite element method and the generalized‐ α method

A velocity‐linked algorithm for solving unsteady fluid–structure interaction (FSI) problems in a fully coupled manner is developed using the arbitrary Lagrangian–Eulerian method. The P2/P1 finite element is used to spatially discretize the incompressible Navier–Stokes equations and structural equations, and the generalized‐ α method is adopted for temporal discretization. Common velocity variables are employed at the fluid–structure interface for the strong coupling of both equations. Because of the velocity‐linked formulation, kinematic compatibility is automatically satisfied and forcing terms do not need to be calculated explicitly. Both the numerical stability and the convergence characteristics of an iterative solver for the coupled algorithm are investigated by solving the FSI problem of flexible tube flows. It is noteworthy that the generalized‐ α method with small damping is free from unstable velocity fields. However, the convergence characteristics of the coupled system deteriorate greatly for certain Poisson's ratios so that direct solvers are essential for these cases. Furthermore, the proposed method is shown to clearly display the advantage of considering FSI in the simulation of flexible tube flows, while enabling much larger time‐steps than those adopted in some previous studies. This is possible through the strong coupling of the fluid and structural equations by employing common primitive variables. Copyright © 2012 John Wiley & Sons, Ltd.     

A coupled symmetric BE–FE method for acoustic fluid–structure interaction

A coupled symmetric BE–FE method for the calculation of linear acoustic fluid–structure interaction in time and frequency domain is presented. In the coupling formulation a newly developed hybrid boundary element method (HBEM) will be used to describe the behaviour of the compressible fluid. The HBEM is based on Hamilton's principle formulated with the velocity potential. The state variables are separated into boundary variables which are approximated by piecewise polynomial functions and domain variables which are approximated by a superposition of static fundamental solutions. The domain integrals are eliminated, respectively, replaced by boundary integrals and a boundary element formulation with a symmetric mass and stiffness matrix is obtained as result. The structure is discretized by FEM. The coupling conditions fulfil C¹-continuity on the interface. The coupled formulation can also be used for eigenfrequency analyses by transforming it from time domain into frequency domain.     

Eigenfrequency bounds for helical MHD flow

Circle theorems are obtained which furnish bounds on the complex eigenfrequencies arising in the linear stability analysis of helical MHD flow. Results for the helical flow of an inviscid, compressible fluid are contained as a special case.     

流固耦合问题的PD-SPH建模与分析

针对涉及结构变形破坏的流固耦合(Fluid-structure interaction, FSI)问题,提出一种基于虚粒子和排斥力的近场动力学(Peridynamics, PD)-光滑粒子动力学(Smoothed particle hydrodynamics, SPH)耦合方法。结合PD方法求解不连续问题以及SPH方法在流体模拟方面的优势,分别采用PD方法与SPH方法求解固体域和流体域,并通过流体粒子-虚粒子接触算法处理流-固界面,既能利用粒子间排斥力有效防止粒子穿透现象发生,又能利用虚粒子修正流体粒子的边界缺陷,提高计算精度。采用PD-SPH耦合方法模拟静水压力作用下的铝板变形问题以及溃坝水流冲击弹性板问题,所得结果与解析解或其它数值结果吻合良好,验证了耦合方法的可行性和有效性。进一步应用耦合方法模拟了流体作用下的结构变形、破坏以及破坏后部分结构运动全过程,验证了PD-SPH耦合方法在流固耦合-结构破坏问题模拟方面的适用性。     

Absorbing boundary conditions for a 3D non-Newtonian fluid-structure interaction model for blood flow in arteries

Blood flow in arteries is characterized by pulse pressure waves due to the interaction with the vessel walls. A 3D fluid-structure interaction (FSI) model in a compliant vessel is used to represent the pressure wave propagation. The 3D fluid is described through a shear-thinning generalized Newtonian model and the structure by a nonlinear hyperelastic model. In order to cope with the spurious reflections due to the truncation of the computational domain, several absorbing boundary conditions are analyzed. First, a 1D hyperbolic model that effectively captures the wave propagation nature of blood flow in arteries is coupled with the 3D FSI model. Extending previous results, an energy estimate is derived for the 3D FSI-1D coupling in the case of generalized Newtonian models. Secondly, absorbing boundary conditions obtained from the 1D model are imposed directly on the outflow sections of the 3D FSI model, and numerical results comparing the different absorbing conditions in an idealized vessel are presented. Results in a human carotid bifurcation reconstructed from medical images are also provided in order to show that the proposed methodology can be applied to anatomically realistic geometries.     

Numerical Study of a Transonic Aircraft Wing for the Prediction of Flutter Failure

In the present paper, computational analysis has been carried out to assess the coupled fluid–structure interaction using NASTRAN finite element approach. A straight swept wing of aluminum material is studied at transonic zone. Analysis has been carried out to find the natural frequency by fluid–structure interaction, then adopting its natural frequency to calculate the reduced frequency for analyzing the flutter effectiveness. A typical case study of plate has been carried out for better understanding the flutter which was then adopted for the swept wing. A fluid–structure interaction phenomenon provides an additional energy to the moving object in terms of frequency in transonic zone. In this speed zone, the divergence speed results a drag that leads to the object to be in a stronger twisting mode resulting in catastrophic failure of the aircraft. The study has defined the flutter boundary of the wing in terms of velocity and frequency which will be very useful in preventing the flutter failure of the aircraft wing through appropriate design improvement or through restriction operational regime.     

Computational study of shock‐wave interaction with solid obstacles using immersed boundary methods

In this study, an immersed boundary (IB) method based on a direct forcing is coupled with a high‐order weighted‐essentially non‐oscillatory (WENO) scheme to simulate fluid–solid interaction (FSI) problems with complex geometries. The IB is a general simulation method for FSI, whereas the WENO is an efficient scheme for fluid flow simulations and shock waves, and both of them work on regular cartesian grids. The effectiveness and the accuracy of the coupled scheme are first analyzed on well‐documented supersonic test problems for a wide range of Mach numbers. The results are in good agreement with both analytical and experimental data. A comprehensive analysis of the interaction of the moving shock through an array of cylinder matrix is then conducted by varying the number of cylinders in the matrix block while keeping the same opening passage. The relaxation length between two adjacent columns of cylinders is kept identical to study uniquely the effect of surface‐to‐volume ratio of the obstacle matrix. It is shown that the configuration with higher surface‐to‐volume ratio produces more post‐shock flow instabilities downstream of the matrix block. The complex shock/shock and shock/vortex interactions are well resolved by the present computation. It is being observed that after the passage of the shock through the cylinder matrix, eddies of different length scales are generated, but the later stage of shock/vortex and shocklet/vortexlet interactions are different for the two cases. The analysis of the PSD of the total kinetic energy globally conforms to Richardson's inviscid cascade. An intermittent peaked PDF of downstream instantaneous vorticity field is obtained in the limit of Re → ∞ . The baroclinic production of vorticity is found to be feeble as previously founded by Sun and Takayama (J. Fluid Mech. 2003; 478 :237–256). Copyright © 2011 John Wiley & Sons, Ltd.     

On the computation of steady‐state compressible flows using a discontinuous Galerkin method

Computation of compressible steady‐state flows using a high‐order discontinuous Galerkin finite element method is presented in this paper. An accurate representation of the boundary normals based on the definition of the geometries is used for imposing solid wall boundary conditions for curved geometries. Particular attention is given to the impact and importance of slope limiters on the solution accuracy for flows with strong discontinuities. A physics‐based shock detector is introduced to effectively make a distinction between a smooth extremum and a shock wave. A recently developed, fast, low‐storage p‐multigrid method is used for solving the governing compressible Euler equations to obtain steady‐state solutions. The method is applied to compute a variety of compressible flow problems on unstructured grids. Numerical experiments for a wide range of flow conditions in both 2D and 3D configurations are presented to demonstrate the accuracy of the developed discontinuous Galerkin method for computing compressible steady‐state flows. Copyright © 2007 John Wiley & Sons, Ltd.     

The Global Nonlinear Galerkin Method for the Solution of von Karman Nonlinear Plate Equations: An Optimal & Faster Iterative Method for the Direct Solution of Nonlinear Algebraic Equations F(x) = 0, using x<sup style='margin-left:-6.5px'>·</sup> = λ[αF + (1 - α)B^TF]

The application of the Galerkin method, using global trial functions which satisfy the boundary conditions, to nonlinear partial differential equations such as those in the von Karman nonlinear plate theory, is well-known. Such an approach using trial function expansions involving multiple basis functions, leads to a highly coupled system of nonlinear algebraic equations (NAEs). The derivation of such a system of NAEs and their direct solutions have hitherto been considered to be formidable tasks. Thus, research in the last 40 years has been focused mainly on the use of local trial functions and the Galerkin method, applied to the piecewise linear system of partial differential equations in the updated or total Lagrangean reference frames. This leads to the so-called tangent-stiffness finite element method. The piecewise linear tangent-stiffness finite element equations are usually solved by an iterative Newton-Raphson method, which involves the inversion of the tangent-stiffness matrix during each iteration. However, the advent of symbolic computation has made it now much easier to directly derive the coupled system of NAEs using the global Galerkin method. Also, methods to directly solve the NAEs, without inverting the tangent-stiffness matrix during each iteration, and which are faster and better than the Newton method are slowly emerging. In a previous paper [Dai, Paik and Atluri (2011a)], we have presented an exponentially convergent scalar homotopy algorithm to directly solve a large set of NAEs arising out of the application of the global Galerkin method to von Karman plate equations. While the results were highly encouraging, the computation time increases with the increase in the number of NAEs-the number of coupled NAEs solved by Dai, Paik and Atluri (2011a) was of the order of 40. In this paper we present a much improved method of solving a larger system of NAEs, much faster. If F(x) = 0 [F_i(x_j) = 0] is the system of NAEs governing the modal amplitudes x_j [j = 1, 2...N], for large N, we recast the NAEs into a system of nonlinear ODEs: x^· = λ[αF + (1 - α)B^TF], where λ and α are scalars, and B_ij = ∂F_i / ∂x_j. We derive a purely iterative algorithm from this, with optimum value for λ and α being determined by keeping x on a newly defined invariant manifold [Liu and Atluri (2011b)]. Several numerical examples of nonlinear von Karman plates, including the post-buckling behavior of plates with initial imperfections are presented to show that the present algorithms for directly solving the NAEs are several orders of magnitude faster than those in Dai, Paik and Atluri (2011a). This makes the resurgence of simple global Galerkin methods, as alternatives to the finite element method, to directly solve nonlinear structural mechanics problems without piecewise linear formulations, entirely feasible.     

FSI modeling with the DSD/SST method for the fluid and finite difference method for the structure

We present a fluid–structure interaction (FSI) modeling method based on using the deforming-spatial-domain/stabilized space–time (DSD/SST) method for the fluid mechanics part and a finite difference (FD) method for the structural mechanics part. As the structural mechanics model, we focus on the thin-shell model. The fluid mechanics equations with moving boundaries are solved with the DSD/SST method and the thin-shell structural mechanics equation is solved with a FD method, with partitioned coupling between the two parts. The coupling of the DSD/SST and FD solvers makes sure that the boundary conditions on the fluid-structure interface at the end of each time step are matched between the fluid and the structure. A hanging plate in vacuum under gravitational force is performed to validate the structure solver. In addition, a pitching plate in a uniform flow is simulated to validate the FSI solver. The present results are in reasonable agreement with data predicted by other methods.