Multi-objective shape optimization of a hydraulic turbine runner using efficiency,strength and weight criteria

An approach for multi-discipline automatic optimization of the hydraulic turbine runner shape is presented. The approach accounts hydraulic efficiency, mechanical strength and the weight of the runner. In order to effectively control the strength and weight of the runner, a new parameterization of the blade thickness function is suggested. Turbine efficiency is evaluated through numerical solution of Reynolds-averaged Navier-Stokes equations, while the finite element method is used to evaluate the von Mises stress in the runner. An objective function, being the weighted sum of maximal stress and the blade volume, is suggested to account for both the strength and weight of the runner. Multi-objective genetic algorithm is used to solve the optimization problem. The suggested approach has been applied to automatic design of a Francis turbine runner. Series of three-objective optimization runs have been carried out. The obtained results clearly indicate that simultaneous account of stress and weight objectives accompanied by thickness variation allows obtaining high efficiency, light and durable turbine runners.     

High-order curvilinear meshing using a thermo-elastic analogy

With high-order methods becoming increasingly popular in both academia and industry, generating curvilinear meshes that align with the boundaries of complex geometries continues to present a significant challenge. Whereas traditional low-order methods use planar-faced elements, high-order methods introduce curvature into elements that may, if added naively, cause the element to self-intersect. Over the last few years, several curvilinear mesh generation techniques have been designed to tackle this issue, utilizing mesh deformation to move the interior nodes of the mesh in order to accommodate curvature at the boundary. Many of these are based on elastic models, where the mesh is treated as a solid body and deformed according to a linear or non-linear stress tensor. However, such methods typically have no explicit control over the validity of the elements in the resulting mesh. In this article, we present an extension of this elastic formulation, whereby a thermal stress term is introduced to ‘heat’ or ‘cool’ elements as they deform. We outline a proof-of-concept implementation and show that the adoption of a thermo-elastic analogy leads to an additional degree of robustness, by considering examples in both two and three dimensions.     

Array-based,parallel hierarchical mesh refinement algorithms for unstructured meshes

In this paper, we describe an array-based hierarchical mesh refinement capability through uniform refinement of unstructured meshes for efficient solution of PDE’s using finite element methods and multigrid solvers. A multi-degree, multi-dimensional and multi-level framework is designed to generate the nested hierarchies from an initial coarse mesh that can be used for a variety of purposes such as in multigrid solvers/preconditioners, to do solution convergence and verification studies and to improve overall parallel efficiency by decreasing I/O bandwidth requirements (by loading smaller meshes and in-memory refinement). We also describe a high-order boundary reconstruction capability that can be used to project the new points after refinement using high-order approximations instead of linear projection in order to minimize and provide more control on geometrical errors introduced by curved boundaries.The capability is developed under the parallel unstructured mesh framework “Mesh Oriented dAtaBase” (MOAB Tautges et al. (2004)). We describe the underlying data structures and algorithms to generate such hierarchies in parallel and present numerical results for computational efficiency and effect on mesh quality. We also present results to demonstrate the applicability of the developed capability to study convergence properties of different point projection schemes for various mesh hierarchies and to a multigrid finite-element solver for elliptic problems.     

Automated mesh generation for tubular joint stress analysis

A technique for automatically generating finite-element meshes to be used in stress analysis of tubular intersections is described. The strengths and limitations of earlier analytical attempts are discussed. The efforts of the authors to extend a somewhat limited existing scheme for mesh generation are treated in detail. The new capabilities made possible by the coupling of the mesh generator with a general-purpose analysis program are explored. The paper closes by considering several applications of the technique along with its possible impact on tubular joint design practice.     

Surface remeshing with robust high-order reconstruction

Remeshing is an important problem in variety of applications, such as finite element methods and geometry processing. Surface remeshing poses some unique challenges, as it must deliver not only good mesh quality but also good geometric accuracy. For applications such as finite elements with high-order elements (quadratic or cubic elements), the geometry must be preserved to high-order (third-order or higher) accuracy, since low-order accuracy may undermine the convergence of numerical computations. The problem is particularly challenging if the CAD model is not available for the underlying geometry, and is even more so if the surface meshes contain some inverted elements. We describe remeshing strategies that can simultaneously produce high-quality triangular meshes, untangling mildly folded triangles and preserve the geometry to high-order of accuracy. Our approach extends our earlier works on high-order surface reconstruction and mesh optimization by enhancing its robustness with a geometric limiter for under-resolved geometries. We also integrate high-order surface reconstruction with surface mesh adaptation techniques, which alter the number of triangles and nodes. We demonstrate the utilization of our method to meshes for high-order finite elements, biomedical image-based surface meshes, and complex interface meshes in fluid simulations.     

Numerical simulation of fluid added mass effect on a francis turbine runner

In this paper, a numerical simulation to analyze the influence of the surrounding water in a turbine runner has been carried out using finite element method (FEM). First, the sensitivity of the FEM model on the element shape and mesh density has been analysed. Secondly, with the optimized FEM model, the modal behaviour with the runner vibrating in air and in water has been calculated. The added mass effect by comparing the natural frequencies and mode shapes in both cases has been determined.The numerical results obtained have been compared with experimental results available. The comparison shows a good agreement in the natural frequency values and in the mode shapes. The added mass effect due to the fluid structure interaction has been discussed in detail.Finally, the added mass effect on the submerged runner is quantified using a non-dimensional parameter so that the results can be extrapolated to runners with geometrical similarity.     

Computer modeling for the coupled stress-wave and structural response

The mesh density and the time-step size requirements of a Finite Element Model used to predict the coupled stress-wave and the structural response to dynamic loading are discussed. The problem chosen for this study is a long cylindrical shell containing a bonded annular solid core subjected to external uniform radial impulse. This shell-core system represents the case and solid propellant of a solid rocket motor. Depending on the geometry and the mechanical properties of the shell-core system, the coupling between the radial stress-waves in the core and the gross structural breathing mode of the shell-core system could be either strong or weak.

When the time required for a through the thickness round trip of radial stress waves in the core is of the same order of magnitude as the gross structural breathing mode period of the whole system, the coupling between the stress-wave propagation and the structural response is strong. A finer mesh of solid elements (NASTRAN HEXA elements) for modeling the solid core is required to predict correctly the peak hoop strains and stresses in the shell. A coarser mesh gives erroneous results.

On the other hand, when the time required for a through the thickness round trip of radial waves is small compared to the structural breathing mode period of the system, the coupling between the wave propagation and the structural response is weak and the two are decoupled. In this case, a coarser mesh of solid elements gives a good estimate of the peak hoop stresses in the shell. These values are not significantly changed if the mesh is made finer.     

A hybrid-stress finite-element formulation for thick multilayer laminates

A hybrid-stress-based finite-element formulation for thick multilayer laminates is presented and evaluated. Particular attention is given to the through-thickness distributions assumed for both stress and displacement components; high order through-thickness distributions are assumed within each layer in the present formulation. Attention is restricted to cylindrical bending of cross-ply laminates and a two-dimensional plane-strain element is derived. Results obtained for through-thickness distributions of stress and displacement are compared with the elasticity solutions for various numbers of layers and increasing thickness to span ratios. Good correlation between computed and elasticity solutions and the need for the inclusion of high-order through-thickness distributions are shown.     

Reconstructing high-order surfaces for meshing

We consider the problem of reconstructing a high-order surface from a given surface mesh. This problem is important for many meshing operations, such as generating high-order finite elements, mesh refinement, mesh smoothing, and mesh adaptation. We introduce two methods called Weighted Averaging of Local Fittings and Continuous Moving Frames. These methods are both based on weighted least squares polynomial fittings and guarantee C ⁰ continuity. Unlike existing methods for reconstructing surfaces, our methods are applicable to surface meshes composed of triangles and/or quadrilaterals, can achieve third and even higher order accuracy, and have integrated treatments for sharp features. We present the theoretical framework of our methods, their accuracy, continuity, experimental comparisons against other methods, and applications in a number of meshing operations.     

Toward design optimization of a Pelton turbine runner

The objective of the present paper is to propose a strategy to optimize the performance of a Pelton runner based on a parametric model of the bucket geometry, massive particle based numerical simulations and advanced optimization strategies to reduce the dimension of the design problem. The parametric model of the Pelton bucket is based on four bicubic Bézier patches and the number of free parameters is reduced to 21. The numerical simulations are performed using the finite volume particle method, which benefits from a conservative, consistent, arbitrary Lagrangian Eulerian formulation. The resulting design problem is of High-dimension with Expensive Black-box (HEB) performance function. In order to tackle the HEB problem, a preliminary exploration is performed using 2’000 sampled runners geometry provided by a Halton sequence. A cubic multivariate adaptive regression spline surrogate model is built according to the simulated performance of these runners. Moreover, an original clustering approach is proposed to decompose the design problem into four sub-problems of smaller dimensions that can be addressed with more conventional optimization techniques.     

On the design optimization of built-up stiffened steel beams with buckling and frequency constraints

The literature on the structural design optimization of steel-plate girders indicates a need for more refined research studies to obtain optimal designs by formulating and solving the design problem that combines structural sizing and shape parameters in one unified, constrained problem. For this purpose, the structural optimization design problem of stiffened steel-plate girders is formulated with specified loading conditions and constraints on strength and serviceability considerations including limits on fundamental frequency and buckling modes. The finite-element method-based model is used to define the objective function and the structural/geometric response functions, while the geometric domain elements are used to systematically perturb the structural shape during the search for an optimal shape of the structure. The mathematical statement of the gradient-based-design problem is solved for an optimal structural size and shape with buckling and frequency constraints in addition to the traditional strength constraints. The numerical results obtained are compared with results obtained from a less formal ad hoc design procedure, and some conclusions are drawn to emphasize the design benefits obtained from solving the design problem for optimal structural size and shape.     

Adaptive topology optimization of elastoplastic structures

Material topology optimization is applied to determine the basic layout of a structure. The nonlinear structural response, e.g. buckling or plasticity, must be considered in order to generate a reliable design by structural optimization. In the present paper adaptive material topology optimization is extended to elastoplasticity. The objective of the design problem is to maximize the structural ductility which is defined by the integral of the strain energy over a given range of a prescribed displacement. The mass in the design space is prescribed. The design variables are the densities of the finite elements. The optimization problem is solved by a gradient based OC algorithm. An elastoplastic von Mises material with linear, isotropic work-hardening/softening for small strains is used. A geometrically adaptive optimization procedure is applied in order to avoid artificial stress singularities and to increase the numerical efficiency of the optimization process. The geometric parametrization of the design model is adapted during the optimization process. Elastoplastic structural analysis is outlined. An efficient algorithm is introduced to determine the gradient of the ductility with respect to the densities of the finite elements. The overall optimization procedure is presented and verified with design problems for plane stress conditions.     

Polygonal multiresolution topology optimization (PolyMTOP) for structural dynamics

We use versatile polygonal elements along with a multiresolution scheme for topology optimization to achieve computationally efficient and high resolution designs for structural dynamics problems. The multiresolution scheme uses a coarse finite element mesh to perform the analysis, a fine design variable mesh for the optimization and a fine density variable mesh to represent the material distribution. The finite element discretization employs a conforming finite element mesh. The design variable and density discretizations employ either matching or non-matching grids to provide a finer discretization for the density and design variables. Examples are shown for the optimization of structural eigenfrequencies and forced vibration problems.     

注塑模流道系统集成设计技术研究

通过总结注塑模流道系统的设计知识,结合基于特征的设计、专家系统技术和ＣＡＥ分析工具,对流道系统进行完整全面的设计,利用专家系统进行流道和浇口的初始设计,采用ＣＡＥ工具Ｃ－ＭＯＬＤ和评价目标函数,进行流道和浇口的优化设计,并总结流道和浇 形状特征,最终建立流道系统的特征和实体模型,这一方法已较好地应用于构造新一代注塑模设计制造系统中。     

Runner sizing in multiple cavity injection mould by non-dominated sorting genetic algorithm

Runner system is important in the plastic injection moulding as it affects the part quality and the material costs. The layout of the runners for a multiple non-identical cavity mould is geometrically imbalance. Even for a multiple identical cavity mould, the layout can be imbalance due to various reasons. This paper presents an approach to balance the flow by adjusting the runner sizes. Runner size determination is a multiobjective optimisation problem. The non-dominated sorting genetic algorithm is adopted for determining the runner sizes. Multiple objective functions including runner balancing, part quality in terms of warpage and runner volume are incorporated into the algorithm. The moulding conditions affecting the mould cavity filling are also determined due to their sensitivity to runner sizes. This runner sizing approach is suitable for the geometric imbalance mouldings and family mouldings.     

Three-dimensional mesh generation using principles of finite element method

The present work deals with the development of a three-dimensional mesh generation algorithm using the principles of FEM with special emphasis on the computational efficiency and the memory requirement. The algorithm makes use of a basic mesh that defines the total number of elements and nodes. Wavefront technique is used to renumber the nodes in order to reduce the bandwidth. By elastic distortion of the basic mesh, it is redefined to map onto actual geometry to be discretized. Later a finer distribution of mesh is done in the zones of interest to suit the nature of the problem. The same Finite Element code meant for stress analysis is adopted with necessary modifications. The algorithm has been extended to three-dimensional geometries. The current methodology is used to discretize a straight bevel gear and an hourglass worm to study their stress patterns.     

An error analysis and mesh adaptation method for shape design of structural components

A finite element error analysis and mesh adaptation method that can be used for improving analysis accuracy in carrying out shape design of structural components is presented in this paper. The simple error estimator developed by Zienkiewicz is adopted in this study for finite element error analysis, using only post-processing finite element data. The mesh adaptation algorithm implemented in ANSYS is investigated and the difficulties found are discussed. An improved algorithm that utilizes ANSYS POST1 capabilities is proposed and found to be more efficient than the ANSYS algorithm. An example is given to show the efficiency. An interactive mesh adaptation method that utilizes PATRAN meshing and result-displaying capabilities is proposed. This proposed method displays error distribution and stress contour of analysis results using color plots, to help the designer in identifying the critical regions for mesh refinement. Also, it provides guidance for mesh refinement by computing and displaying the desired element size information, based on error estimate and a mesh refinement criterion defined by the designer. This method is more efficient and effective than the semi-automatic algorithm implemented in ANSYS, and is suitable for structural shape design. This method can be applied not only to set-up a finite element mesh of the structure at initial design but to ensure analysis accuracy in the design process. Examples are given to demonstrate feasibility of the proposed method.     

Invisible runners in finite fields

Suppose k runners having different constant speeds run laps on a circular track of unit length. The Lonely Runner Conjecture states that, sooner or later, any given runner is at distance at least 1/k from all the other runners. We prove here that the statement of the conjecture holds if we eliminate only one chosen runner. The proof uses a simple double-counting argument in the setting of finite fields. We also demonstrate that the original problem reduces to an analogous statement in particular ring Zn, where n is the sum of speeds of two distinct runners. In consequence we obtain a simple computational procedure for verifying the conjecture for any given set of integer speeds. Finally we derive some simple consequences of our results for coloring integer distance graphs.     

A Spectral Finite Element Approach to Modeling Soft Solids Excited with High-Frequency Harmonic Loads

An approach for efficient and accurate finite element analysis of harmonically excited soft solids using high-order spectral finite elements is presented and evaluated. The Helmholtz-type equations used to model such systems suffer from additional numerical error known as pollution when excitation frequency becomes high relative to stiffness (i.e. high wave number), which is the case, for example, for soft tissues subject to ultrasound excitations. The use of high-order polynomial elements allows for a reduction in this pollution error, but requires additional consideration to counteract Runge's phenomenon and/or poor linear system conditioning, which has led to the use of spectral element approaches. This work examines in detail the computational benefits and practical applicability of high-order spectral elements for such problems. The spectral elements examined are tensor product elements (i.e. quad or brick elements) of high-order Lagrangian polynomials with non-uniformly distributed Gauss-Lobatto-Legendre nodal points. A shear plane wave example is presented to show the dependence of the accuracy and computational expense of high-order elements on wave number. Then, a convergence study for a viscoelastic acoustic-structure interaction finite element model of an actual ultrasound driven vibroacoustic experiment is shown. The number of degrees of freedom required for a given accuracy level was found to consistently decrease with increasing element order. However, the computationally optimal element order was found to strongly depend on the wave number.     

FE-shape sensitivity of elastoplastic response

A new approach for performing FE-shape design sensitivity analyses (DSA) of structural models with both linear elastic and elastoplastic material behaviour is presented. In the formulation of the method the derivation of the FE equilibrium equations is performed analytically leading to various terms. The differentiation of some parts of these terms is determined numerically, therefore the method is semianalytical. The formulation is particularized for isoparametric finite elements for which exact numerical differentiation can be obtained (exact up to round-off error). Examples of some plane stress problems testify that the results of the new method are not dependent either on the size of perturbation of the design variables or on the FE mesh refinement among other factors.