Shape optimization of an airfoil in a BZT flow with multiple-source uncertainties

Bethe–Zel’dovich–Thompson fluids (BZT) are characterized by negative values of the fundamental derivative of gasdynamics for a range of temperatures and pressures in the vapor phase, which leads to non-classical gasdynamic behaviors such as the disintegration of compression shocks. These non-classical phenomena can be exploited, when using these fluids in Organic Rankine Cycles (ORCs), to increase isentropic efficiency. A predictive numerical simulation of these flows must account for two main sources of physical uncertainties: the BZT fluid properties often difficult to measure accurately and the usually fluctuating turbine inlet conditions. For taking full advantage of the BZT properties, the turbine geometry must also be specifically designed, keeping in mind the geometry achieved in practice after machining always slightly differs from the theoretical shape. This paper investigates some efficient procedures to perform shape optimization in a 2D BZT flow with multiple-source uncertainties (thermodynamic model, operating conditions and geometry). To demonstrate the feasibility of the proposed efficient strategies for shape optimization in the presence of multiple-source uncertainties, a zero incidence symmetric airfoil wave-drag minimization problem is retained as a case-study. This simplified configuration encompasses most of the features associated with a turbine design problem, as far the uncertainty quantification is concerned. A preliminary analysis of the contributions to the variance of the wave-drag allows to select the most significant sources of uncertainties using a reduced number of flow computations. The resulting mean value and variance of the objective are next turned into metamodels. The optimal Pareto sets corresponding to the minimization of various substitute functions are obtained using a genetic algorithm as optimizer and their differences are discussed.     

Gridless spectral algorithm for Stokes flow with moving boundaries

A gridless, spectrally-accurate algorithm for the Stokes flow with moving boundaries is presented. The algorithm uses fixed computational domain with boundaries of the flow domain moving inside the computational domain. The spatial discretization is based on the Fourier expansions in the streamwise direction and the Chebyshev expansions in the transverse direction. Temporal discretization uses one- and two-steps implicit formulations. The boundary conditions on the moving boundaries are imposed using the immersed boundary conditions concept. Numerical tests confirm the spectral accuracy in space and theoretically-predicted accuracy in time. Different variants of the solution procedure are presented and their relative advantages are discussed.     

Finite element methods for flow problems with moving boundaries and interfaces

Summary  This paper is an overview of the finite element methods developed by the Team for Advanced Flow Simulation and Modeling (T*AFSM) [http://www.mems.rice.edu/TAFSM/] for computation of flow problems with moving boundaries and interfaces. This class of problems include those with free surfaces, two-fluid interfaces, fluid-object and fluid-structure interactions, and moving mechanical components. The methods developed can be classified into two main categories. The interface-tracking methods are based on the Deforming-Spatial-Domain/Stabilized Space-Time (DSD/SST) formulation, where the mesh moves to track the interface, with special attention paid to reducing the frequency of remeshing. The interface-capturing methods, typically used for free-surface and two-fluid flows, are based on the stabilized formulation, over non-moving meshes, of both the flow equations and the advection equation governing the time-evolution of an interface function marking the location of the interface. In this category, when it becomes neccessary to increase the accuracy in representing the interface beyond the accuracy provided by the existing mesh resolution around the interface, the Enhanced-Discretization Interface-Capturing Technique (EDICT) can be used to to accomplish that goal. In development of these two classes of methods, we had to keep in mind the requirement that the methods need to be applicable to 3D problems with complex geometries and that the associated large-scale computations need to be carried out on parallel computing platforms. Therefore our parallel implementations of these methods are based on unstructured grids and on both the distributed and shared memory parallel computing approaches. In addition to these two main classes of methods, a number of other ideas and methods have been developed to increase the scope and accuracy of these two classes of methods. The review of all these methods in our presentation here is supplemented by a number numerical examples from parallel computation of complex, 3D flow problems.     

Unsteady flow computations for flow past multiple moving boundaries using LSKUM

We present the latest developments in the least squares kinetic upwind method (LSKUM), a kinetic theory based grid free approach for the solution of Euler equations. A single step higher order scheme through modified CIR splitting is presented. A new weighted least squares method has been used in the present work which simplifies the 2-D formulae to an equivalent 1-D form. This is achieved through diagonalisation of the least squares matrix through suitable choices of the weights. All these developments have been extended to problems with moving nodes and boundaries. A 2-D unsteady Euler code has been developed incorporating all the above ideas along with the well known dual time stepping procedure. The code has been verified and validated for the standard test case AGARD CT(5) which corresponds to unsteady flow past oscillating NACA0012 airfoil pitching about quarter chord. Good comparisons with the experimental values have been obtained. In order to demonstrate the ability of the method to handle multiple moving bodies we have computed unsteady flow past two oscillating NACA0012 airfoils one behind the other. Some interesting results are presented for this case.     

Laminar viscous flow fields in moving curved channels

The laminar steady flow of air through moving narrow deep circumferential grooves on a rotor is studied. Fluid enters the moving grooves from a stationary inlet nozzle with uniform velocity and exits into a stationary diffuser after approximately 150 degrees of rotation. Three dimensional developing velocity profiles and pressure distributions are presented for a variety of operating conditions. Computed overall head-flow characteristics are found to be in relatively close agreement with a simple Poiseuille model where both walls are taken to be moving at the same rate and in the same direction at a velocity different from the entrance velocity.     

Shape design sensitivity analysis of kinematical boundaries

A new approach is used in this paper to derive the design sensitivity formulation with kinematical design boundaries. By employing the concept of the conventional finite difference approach, the variation of structural response due to change of the kinematic design boundary can be represented by the perturbed structure under a set of kinematical boundary conditions. Parameterization of the design variation with respect to the design variable enables us to transform the design sensitivity into the solutions of a boundary value problem with perturbation displacements on the design boundary. The perturbation diplacements can be evaluated from the stress and displacement fields of the initial problem. This approach can be treated as a special case of the general direct formulation, but the derivation using the finite difference procedure gives a strong physical meaning of the method, and the formulation derived provides an explicit form for design sensitivity calculation. The numerical implementation of this approach based on the boundary element method is discussed, and a few numerical examples are used to verify the proposed formulation.     

Computation of MHD flow due to moving boundaries

A non-iterative numerical scheme is presented which computes in a single iteration the steady, laminar flow of a viscous, incompressible, electrically conducting fluid caused by moving boundaries in the presence of a transverse magnetic field. It also eliminates the possible error induced by taking the value of numerical infinity (representing the unbounded domain of the flow) as a finite number. The scheme is based on implicit use of infinite series of exponentials for velocity components. The issue of convergence of these series is also discussed. An asymptotic solution valid for large values of M, the Hartmann number, and an approximate solution valid for any value of M are further developed. In particular, the case of axisymmetric magnetohydrodynamic (MHD) flow due to a stretching sheet has been dealt with in some detail. A comparison has been made of the merits of various techniques used in the paper and appropriate conclusions are drawn.     

Topological derivatives applied to fluid flow channel design optimization problems

Topology optimization methods application for viscous flow problems is currently an active area of research. A general approach to deal with shape and topology optimization design is based on the topological derivative. This relatively new concept represents the first term of the asymptotic expansion of a given shape functional with respect to the small parameter which measures the size of singular domain perturbations, such as holes and inclusions. In previous topological derivative-based formulations for viscous fluid flow problems, the topology is obtained by nucleating and removing holes in the fluid domain which creates numerical difficulties to deal with the boundary conditions for these holes. Thus, we propose a topological derivative formulation for fluid flow channel design based on the concept of traditional topology optimization formulations in which solid or fluid material is distributed at each point of the domain to optimize the cost function subjected to some constraints. By using this idea, the problem of dealing with the hole boundary conditions during the optimization process is solved because the asymptotic expansion is performed with respect to the nucleation of inclusions – which mimic solid or fluid phases – instead of inserting or removing holes in the fluid domain, which allows for working in a fixed computational domain. To evaluate the formulation, an optimization problem which consists in minimizing the energy dissipation in fluid flow channels is implemented. Results from considering Stokes and Navier-Stokes are presented and compared, as well as two- (2D) and three-dimensional (3D) designs. The topologies can be obtained in a few iterations with well defined boundaries.     

Topology optimization for the design of flow fields in a redox flow battery

This paper presents topology optimization for the design of flow fields in vanadium redox flow batteries (VRFBs), which are large-scale storage systems for renewable energy resources such as solar and wind power. It is widely known that, in recent VRFB systems, one of the key factors in boosting charging or discharging efficiency is the design of the flow field around carbon fiber electrodes and in flow channels. In this study, topology optimization is applied in order to achieve optimized flow field designs. The optimization problem is formulated as a maximization problem for the generation rate of the vanadium species governed by a simplified electrochemical reaction model. A typical porous model is incorporated into the optimization problem for expressing the carbon fiber electrode; furthermore, a mass transfer coefficient that depends on local velocity is introduced. We investigate the dependencies of the optimized configuration with respect to the porosity of the porous electrode and the pressure loss. Results indicate that patterns of interdigitated flow fields are valid designs for VRFBs.     

Topology optimization of channel flow problems

This paper describes a topology design method for simple two-dimensional flow problems. We consider steady, incompressible laminar viscous flows at low-to-moderate Reynolds numbers. This makes the flow problem nonlinear and hence a nontrivial extension of the work of Borrvall and Petersson (2003).Further, the inclusion of inertia effects significantly alters the physics, enabling solutions of new classes of optimization problems, such as velocity-driven switches, that are not addressed by the earlier method. Specifically, we determine optimal layouts of channel flows that extremize a cost function which measures either some local aspect of the velocity field or a global quantity, such as the rate of energy dissipation. We use the finite element method to model the flow, and we solve the optimization problem with a gradient-based math-programming algorithm that is driven by analytical sensitivities. Our target application is optimal layout design of channels in fluid network systems. Using concepts borrowed from topology optimization of compliant mechanisms in solid mechanics, we introduce a method for the synthesis of fluidic components, such as switches, diodes, etc.     

Reliability-based design optimization using a family of methods of moving asymptotes

In this study, an effective method for reliability-based design optimization is proposed enhancing sequential optimization and reliability assessment (SORA) method by a family of methods of moving asymptotes (MMA) approximations. In SORA, reliability estimation and deterministic optimization are performed sequentially. And the sensitivity and function value of probabilistic constraint at the most probable point (MPP) are obtained in the process of finding reliability information. In this study, a family of MMA approximations are constructed by utilizing the sensitivity and function value of the probabilistic constraint at the MPP. So, no additional evaluation of the probabilistic constraint is required in constructing MMA approximations. Moreover, no additional evaluation of the probabilistic constraint is required in the deterministic optimization of SORA by using a family of MMA approximations. The efficiency and accuracy of the proposed method were verified through numerical examples.     

一类活动边界分布参数系统的移动控制策略

针对一类具有活动边界的分布参数系统,考虑移动传感器/执行器的动力学行为,研究其移动控制问题.考虑并列的传感器/执行器,通过传感器对系统的状态进行测量并将测量值传递给执行装置中的控制器.根据传感器之间不同的信息传递方式分别设计相应的控制器,基于无穷维抽象发展方程理论和Lyapunov方法得到相应的移动控制策略,并证明所提出的移动控制策略与系统的活动边界之间的关系.最后通过仿真结果验证所提出策略的有效性.     

Shape optimization for a link mechanism

This paper presents a numerical solution for shape optimization problems for link mechanisms, such as a piston-crank mechanism. The dynamic behavior of a link mechanism is described by a differential-algebraic equation (DAE) system consisting of motion equations for each single body and constraints of linkages and rigid motions. In a shape optimization problem, the objective function to maximize is constructed from the external work done by a given external force, which agrees with the kinetic energy of the link mechanism, for an assigned time interval, and the total volume of all the links forms the constraint function. The Fréchet derivatives of these cost functions with respect to the domain variation, which we call the shape derivatives of these cost functions, are evaluated theoretically. A scheme to solve the shape optimization problem is presented using the H ¹ gradient method (the traction method) proposed by the authors as a reshaping algorithm, since it retains the smoothness of the boundary. A numerical example shows that reasonable shapes for each link such that mobility of the link mechanism is improved are obtained by this approach.     

Filtering in distributed parameter systems with moving boundaries

A general filter is presented for nonlinear distributed parameter systems having an arbitrary number of moving boundaries. The filter is capable of including both volume and boundary process noise as well as measurement errors. A large number of potential applications of the results, ranging from the estimation of the extent of crude oil spills to metallurgical solidification problems, are discussed.     

Numerical analysis of a 2-D viscous sintering problem with non-smooth boundaries

By viscous sintering it is meant the process of bringing a granular compact to a temperature at which the viscosity of the material becomes low enough for surface tension to cause the particles to deform and coalesce, whereby the material transport can be modelled as a viscous incompressible newtonian volume flow. Here a two-dimensional model is considered. A Boundary Element Method is applied to solve the governing Stokes creeping flow equations for an arbitrarily initial shaped fluid region. In this paper we show that the viscous sintering problem is well-conditioned from an evolutionary point of view. However as boundary value problem at each time step, the problem is ill-conditioned when the contact surfaces of the particles are small, i.e. in the early stages of the coalescence. This is because the curvature of the boundary at those places can be very large. This ill-conditioning is illustrated by an example: the coalescence of two equal circles. This example demonstrates the main evolutionary features of the sintering phenomenon very well. A numerical consequence of this ill-conditioning is that special care has to be taken for distributing and redistributing the nodal points at these boundary parts. Therefore an algorithm for this node redistribution is outlined. Several numerical examples sustain the analysis.     

Estimating shear stress in a rectangular channel with rough boundaries using an optimized SVM method

Neural Computing and Applications - The accurate simulation of wall and bed shear stresses in rectangular channels is one of the most important topics in hydraulic engineering. In this study, the...     

Shape preserving design of vibrating structures using topology optimization

In several engineering components, the shape of some functional surfaces needs to be preserved in order to avoid losing performance or even its functionality when subjected to loads. This is particularly important when tight tolerances are required for operational conditions in some regions. If the deformation significantly affects product functionality, it is interesting to use a shape preserving design technique. This will often reduce deformation in a local region. To achieve that, we deal with topology optimization of elastic, continuum structures with Rayleigh damping, subjected to time-harmonic, design-independent external dynamic loading with prescribed excitation frequency, amplitude and spatial distribution. In topology optimization for vibrating structures, the obtained design should often have its resonance frequencies driven far away from the given excitation frequency in order to avoid resonance and to reduce vibration levels. In this work, we explore harmonic vibration problems with the excitation frequency lower than the first resonance frequency of the initial structure. Dynamic compliance minimization is used to improve dynamic response of the structure. An additional local dynamic compliance constraint is used to define the shape preserving problem, thus, reducing deformation in specific regions of a part named shape preserving region (SPR). A commercial FE code (ANSYS^?) is used to solve the finite element problem. The optimization Method of Moving Asymptotes (MMA) is used with the modified Solid Isotropic Material with Penalization (SIMP) material interpolation scheme. The effectiveness of this technique is presented using 2D plane structures. Coherent results were achieved using the proposed optimization formulation. It is possible to observe significant decrease on local deformation, at expense of little increase on global dynamic compliance.