High-fidelity aerostructural optimization with integrated geometry parameterization and mesh movement

This paper extends an integrated geometry parameterization and mesh movement strategy for aerodynamic shape optimization to high-fidelity aerostructural optimization based on steady analysis. This approach provides an analytical geometry representation while enabling efficient mesh movement even for very large shape changes, thus facilitating efficient and robust aerostructural optimization. The geometry parameterization methodology uses B-spline surface patches to describe the undeflected design and flying shapes with a compact yet flexible set of parameters. The geometries represented are therefore independent of the mesh used for the flow analysis, which is an important advantage to this approach. The geometry parameterization is integrated with an efficient and robust grid movement algorithm which operates on a set of B-spline volumes that parameterize and control the flow grid. A simple technique is introduced to translate the shape changes described by the geometry parameterization to the internal structure. A three-field formulation of the discrete aerostructural residual is adopted, coupling the mesh movement equations with the discretized three-dimensional inviscid flow equations, as well as a linear structural analysis. Gradients needed for optimization are computed with a three-field coupled adjoint approach. Capabilities of the framework are demonstrated via a number of applications involving substantial geometric changes.     

Equi-affine Invariant Geometry for Shape Analysis

Traditional models of bendable surfaces are based on the exact or approximate invariance to deformations that do not tear or stretch the shape, leaving intact an intrinsic geometry associated with it. These geometries are typically defined using either the shortest path length (geodesic distance), or properties of heat diffusion (diffusion distance) on the surface. Both measures are implicitly derived from the metric induced by the ambient Euclidean space. In this paper, we depart from this restrictive assumption by observing that a different choice of the metric results in a richer set of geometric invariants. We apply equi-affine geometry for analyzing arbitrary shapes with positive Gaussian curvature. The potential of the proposed framework is explored in a range of applications such as shape matching and retrieval, symmetry detection, and computation of Voroni tessellation. We show that in some shape analysis tasks, equi-affine-invariant intrinsic geometries often outperform their Euclidean-based counterparts. We further explore the potential of this metric in facial anthropometry of newborns. We show that intrinsic properties of this homogeneous group are better captured using the equi-affine metric.     

The LIR space partitioning system applied to the Stokes equations

We introduce a novel approach to solve the stationary Stokes equations on very large voxel geometries. One main idea is to coarsen a voxel geometry in areas where velocity does not vary much while keeping the original resolution near the solid surfaces. For spatial partitioning a simplified LIR-tree is used which is a generalization of the Octree and KD-tree. The other main idea is to arrange variables in a way such that each cell is able to satisfy the Stokes equations independently. Pressure and velocity are discretized on staggered grids. But instead of using one velocity variable on the cell surface we introduce two variables. The discretization of momentum and mass conservation yields a small linear system (block) per cell that allows to use the block Gauss–Seidel algorithm as iterative solver. We compare our method to other solvers and conclude superior performance in runtime and memory for high porosity geometries.     

Pseudospectral matrix element methods for flow in complex geometry

A pseudospectral matrix element (PSME) method, which extended the global pseudospectral method to a multi-element scheme, has been applied to the solution of the incompressible, primitive variable, Navier-Stokes equations for complex geometries with rectilinear or curvilinear boundaries. For a simple complex geometry, a direct solution for pressure Poisson equation is feasible, while in a much more complex geometry the pressure solution is accomplished by a new implementation of domain decomposition approach. According to this approach, the computational domain can be divided into a number of overlapping subdomains where the grid points inside the overlapping area may or may not be located at the same place. Each subdomain can be mapped onto a square domain by an algebraic (or isoparametric) mapping, of simpler geometry with patched elements, in which the pressure solution is more easily obtained by an eigenfunction expansion technique for cartesian-type geometries or a direct solver for noncartesian-type geometries with rectilinear (or curvilinear) boundaries. With an iterative Schwarz alternating procedure (SAP) between subdomains, the complete solution is found. The novel feature of this approach are (i) the continuity equation is satisfied everywhere, in the interior (including the inter-element points) and on the boundary; (ii) reducing the global storage size to local (subdomain) storage locations for which parallel computation is easily implemented; (iii) producing the desired grid points without solving any grid-generating equations is easy; and (iv) consistent mass conservation holds at geometrical singular points despite their discontinuous slope (i.e., singular vorticity). Numerical examples of flow over a triangular and parabolic bump as well as flow in a bifurcation with a daughter branch entering the main channel at angles 45° and 90° are presented in this paper.     

Variational geometry: A new method for modifying part geometry for finite element analysis

Interactive graphic methods have the potential to significantly reduce the cost associated with pre- and post-processing of finite element analyses. One area of particular importance is the creation and modification of part geometry.

This paper describes a powerful method for modification of geometry for finite element analysis pre-processors. The method, called “Variational Geometry”, uses a single representation to describe the entire family of geometries that share a generic shape.

A solid geometric model of a component is defined with respect to a set of scalar parameters. Dimensions, such as those which appear on a mechanical drawing, are treated as constraints on the permissible values of these parameters. Constraints on the geometry are expressed as a set of non-linear algebraic equations. The values of the parameters and hence the geometry may be determined by solving the set of non-linear constraint equations.

A procedure for minimizing the computational requirements is presented. For a part with n degrees of freedom, the solution time is shown to be O(n).     

Isogeometric shape optimization of photonic crystals via Coons patches

In this paper, we present an approach that extends isogeometric shape optimization from optimization of rectangular-like NURBS patches to the optimization of topologically complex geometries. We have successfully applied this approach in designing photonic crystals where complex geometries have been optimized to maximize the band gaps.Salient features of this approach include the following: (1) multi-patch Coons representation of design geometry. The design geometry is represented as a collection of Coons patches where the four boundaries of each patch are represented as NURBS curves. The use of multiple patches is motivated by the need for representing topologically complex geometries. The Coons patches are used as a design representation so that designers do not need to specify interior control points and they provide a mechanism to compute analytical sensitivities for internal nodes in shape optimization, (2) exact boundary conversion to the analysis geometry with guaranteed mesh injectivity. The analysis geometry is a collection of NURBS patches that are converted from the multi-patch Coons representation with geometric exactness in patch boundaries. The internal NURBS control points are embedded in the parametric domain of the Coons patches with a built-in mesh rectifier to ensure the injectivity of the resulting B-spline geometry, i.e. every point in the physical domain is mapped to one point in the parametric domain, (3) analytical sensitivities. Sensitivities of objective functions and constraints with respect to design variables are derived through nodal sensitivities. The nodal sensitivities for the boundary control points are directly determined by the design parameters and those for internal nodes are obtained via the corresponding Coons patches.     

Optimization of folded plate roofs

Increased usage of prefabricated structural components and the advantages offered by folded plate roofs necessitates the optimization of their configurations. In order to reduce the material expenditure, transportation and erection costs the dead-weight of the roof systems need to be minimized.The reported study defines a technique for the optimization of the structural weight of the folded plate roof without intermediate stiffeners and subjected to one loading system. The developed method enabled the generation of the different, and consequently the optimum, geometries by starting with an arbitrary initial geometry. The azimuthal angles, the width and the thickness of the individual panels were permitted to vary. The optimization employed a variational approach and flexibility formulation. The total weight of the roof was taken as the target function, subject to minimization. The equilibrium equations for the panels were taken as the constraint equations for the optimization process. The pseudo-weight function was established through the use of the Lagrangian multipliers. The standard Lagrangian formulation was then applied to the pseudo-weight function, and a set of optimization equations were generated. These equations, along with the original equilibrium equations, formed a system of simultaneous nonlinear transcendental equations. This set was solved using an iterative approach.The overall formulation was kept general enough to permit the inclusion of any given loading condition to permit the application of the given methodology to any folded plate roof system.     

Transcendental functions and mechanical theorem proving in elementary geometries

We present a new approach to dealing with certain problems about trigonometric and hyperbolic function using Wu's method by transforming the two kinds of transcendental functions into polynomials. This approach has been used for mechanical theorem proving in Euclidean and several Non-Euclidean geometries. Based on our approach, several meta-theorems are established for four Non-Euclidean geometries. These meta-theorems assert that certain kinds of geometry statements that can be confirmed by our approach in one geometry can also be confirmed in three other geometries. We also proved a meta-theorem which permits us to obtain all hyperbolic identities from trigonometric identities through certain kinds of transformation and vice versa.The revision was supported by the NSF grant CCR-8702108.     

Some algorithms to correct a geometry in order to create a finite element mesh

One of the major difficulties in meshing 3D complex geometries is to deal with non-proper geometrical definitions coming from CAD systems. Typically, CAD systems do not take care of the proper definition of the geometries for the analysis purposes. In addition, the use of standard CAD files (IGES, VDA, …) for the transfer of geometries between different systems introduce some additional difficulties.In this work, a collection of algorithms to repair and/or to improve the geometry definitions are provided. The aim of these algorithms is to make as easy as possible the generation of a mesh over complex geometries given some minimum requirements of quality and correctness. The geometrical model will be considered as composed of a set of NURBS lines and trimmed surfaces.Some examples of application of the algorithms and of the meshes generated from the corrected geometry are also presented in this work.     

Computer-aided design of porous artifacts

Heterogeneous structures represent an important new frontier for 21st century engineering. Human tissues, composites, ‘smart’ and multi-material objects are all physically manifest in the world as three-dimensional (3D) objects with varying surface, internal and volumetric properties and geometries. For instance, a tissue engineered structure, such as bone scaffold for guided tissue regeneration, can be described as a heterogeneous structure consisting of 3D extra-cellular matrices (made from biodegradable material) and seeded donor cells and/or growth factors.The design and fabrication of such heterogeneous structures requires new techniques for solid models to represent 3D heterogeneous objects with complex material properties. This paper presents a representation of model density and porosity based on stochastic geometry. While density has been previously studied in the solid modeling literature, porosity is a relatively new problem. Modeling porosity of bio-materials is critical for developing replacement bone tissues. The paper uses this representation to develop an approach to modeling of porous, heterogeneous materials and provides experimental data to validate the approach. The authors believe that their approach introduces ideas from the stochastic geometry literature to a new set of engineering problems. It is hoped that this paper stimulates researchers to find new opportunities that extend these ideas to be more broadly applicable for other computational geometry, graphics and computer-aided design problems.     

Coupled urban wind flow and indoor natural ventilation modelling on a high-resolution grid: A case study for the Amsterdam ArenA stadium

Wind flow in urban environments is an important factor governing the dispersion of heat and pollutants from streets, squares and buildings. This paper presents a coupled CFD modelling approach for urban wind flow and indoor natural ventilation. A specific procedure is used to efficiently and simultaneously generate the geometry and the high-resolution body-fitted grid for both the outdoor and indoor environment. This procedure allows modelling complex geometries with full control over grid quality and grid resolution, contrary to standard semi-automatic unstructured grid generation procedures. It also provides a way to easily implement various changes in the model geometry and grid for parametric studies. As a case study, a parametric analysis of natural ventilation is performed for the geometrically complex Amsterdam ArenA stadium in the Netherlands. The turbulent wind flow and temperature distribution around and inside the stadium are solved with the 3D steady Reynolds-averaged Navier–Stokes equations. Special attention is given to CFD solution verification and validation. It is shown that small geometrical modifications can increase the ventilation rate by up to 43%. The coupled modelling approach and grid generation procedure presented in this paper can be used similarly for future studies of wind flow and related processes in complex urban environments.     

Geometry optimization within a modified extended Hückel formalism: Modifications to the ASED program

The ASED program by Anderson and coworkers was modified to include a full geometry optimizer. It was discovered that, even with this modification, the ASED program could not be used to perform geometry optimizations on organic molecules without further modification. These modifications included changing the Wolfsberg-Helmholz equation used in the ASED program back to the original equation found in the FORTICON8 program as well as changing the Slater orbital exponents. A general parameter set for C, H, O and N has been developed, which has been used to perform geometry optimizations on selected organic molecules. The calculated geometries compared reasonably with the experimental values and with previously published AM1 geometries.     

On the kinematics of multiple manipulator space free-flyers and their computation

In this article, two basic approaches for kinematics modelling of multiple manipulator space free-flying robots (SFFRs) are developed. In the barycentric vector approach, the center of mass of the whole system is taken as a representative point for the translational motion of the system, and a set of body-fixed vectors which reflect both geometric configuration and mass distribution of the system are used. On the other hand, the direct path method relies on taking a point on the base body (preferably its center of mass) as the representative point for the translational motion of the system. The consequences of using each of the two approaches in deriving dynamics equations and in control design of SFFRs are discussed. It is revealed that the direct path method is a more appropriate approach for modelling multiple arm systems, in the presence of external forces/torques (i.e., free-flying mode). A 14 degree-of-freedom space free-flying system is considered as a benchmark system and a quantitative comparison between the two approaches is presented. The results show that the direct path method requires significantly less computations for position and velocity analyses. © 1998 John Wiley & Sons, Inc.     

Active,Foveated, Uncalibrated Stereovision

Biological vision systems have inspired and will continue to inspire the development of computer vision systems. One biological tendency that has never been exploited is the symbiotic relationship between foveation and uncalibrated active, binocular vision systems. The primary goal of any binocular vision system is the correspondence of the two retinal images. For calibrated binocular rigs the search for corresponding points can be restricted to epipolar lines. In an uncalibrated system the precise geometry is unknown. However, the set of possible geometries can be restricted to some reasonable range; and consequently, the search for matching points can be confined to regions delineated by the union of all possible epipolar lines over all possible geometries. We call these regions epipolar spaces. The accuracy and complexity of any correspondence algorithm is directly proportional to the size of these epipolar spaces. Consequently, the introduction of a spatially variant foveation strategy that reduces the average area per epipolar space is highly desirable. This paper provides a set of sampling theorems that offer a path for designing foveation strategies that are optimal with respect to average epipolar area.     

Design of maximum thrust plug nozzles with variable inlet geometry

An optimization analysis is presented for axisymmetric plug nozzles with varible inlet geometry. The analysis is based on the governing gas dynamic relations for a rotational flow of a frozen or equilibrium gas mixture. The problem is formulated to maximize the axial thrust produced by the plug nozzle for a general isoperimetric constraint, such as constant nozzle length or constant nozzle surface area. The effects of base pressure and ambient pressure are included in the thrust expression to be maximized. The governing gas dynamic equations and the differential and integral constraints that the solution must satisfy are incorporated into the formulation by means of Lagrange multiples. The formalism of the calculus of variations is applied to the resulting functional to be maximized. The results of the optimization analysis are a set of partial differential equations for determining the Lagrange multipliers in the region of interest and a set of equations for determining the necessary boundary conditions for the solution. The complete set of equations for the gas dynamic properties and the Lagrange multipliers are system of first order, quasi-linear, non-homogeneous partial differential equations of the hyperbolic type, which can be treated by the method of charac- teristics. The characteristic and compatibility equations for the system are presented. A numerical solution procedure is presented to determine wether or not a given plug nozzle geometry is an optimal solution. An iteration technique is developed which systematically adjusts the plug nozzle geometry until the optimal solution is obtained. Selected parametric studies are presented. These studies illustrate the effect of the specific heat ratio, the design pressure ratio and the base pressure model on the thrust peformance and nozzle geometry of optimal, fixed length, plug nozzles.     

基于邻域预测的三角形网格几何信息压缩

在现有的代表性三角形网格压缩方法中,先采用一定的网格遍历方法来压缩连接信息,同时用遍历路径上的相邻顶点来对每个顶点的几何坐标进行平行四边形预测,以压缩几何信息。它们的主要缺点是平行四边形预测不太准确,且受到所采用的遍历方法的制约。文章提出一种新的几何信息压缩方法。编码时,对每个顶点的几何坐标,采用比平行四边形预测更为准确、且与遍历方法无关的邻域预测。解码时,采用预处理共轭梯度法,联立求解所有顶点的预测公式组成的稀疏线性方程组,同时求出所有顶点的坐标。文章采用渐进解码方法来减少求解稀疏线性方程组时,用户的等待时间。     

n-tuple complex helical geometry modeling using parametric equations

Complex helical structures are difficult to model in three-dimensional form to conduct finite element analysis. In general, parametric mathematical equations for single and double helical geometries are readily available in existing literature. However, more complex forms such as triple or in general n-tuple helical structures are still not widely studied. In this paper, at first, definitions of single and double helical structures are presented in parametric mathematical forms. Centerlines, curvatures, and torsions of these geometries are found, and these two helical geometries are visualized in three-dimensional structures. Next, one of the untouched helical models, the triple helical geometry, is investigated and a procedure to find the centerline of the triple helical geometry is presented. In addition, the first three-dimensional generated solid model of a triple helix geometry is presented. Finally, the steps used to create triple helix geometry are generalized to find parametric mathematical equations for n-tuple helical geometries.     

Multi-objective mobile robot path planning problem through learnable evolution model

A new multi-objective non-Darwinian-type evolutionary computation approach based on learnable evolution model (LEM) is proposed for solving the robot path planning problem. The multi-objective property of this approach is governed by a robust strength Pareto evolutionary algorithm (SPEA) incorporated in the LEM algorithm presented here. Learnable evolution model includes a machine learning method, like the decision trees, that can detect the right directions of the evolution and leads to large improvements in the fitness of the individuals. Several new refiner operators are proposed to improve the objectives of the individuals in the evolutionary process. These objectives are: the path length, the path safety and the path smoothness. A modified integer coding path representation scheme is proposed where the edge-fixing and top-row fixing procedures are performed implicitly. This proposed robot path planning problem solving approach is assessed on eight realistic scenarios in order to verify the performance thereof. Computer simulations reveal that this proposed approach exhibits much higher hypervolume and set coverage in comparison with other similar approaches. The experimental results confirm that the proposed approach performs in the workspaces with a dense set of obstacles in a significant manner.