On Simulated Annealing and the Construction of Linear Spline Approximations for Scattered Data

Abstract. We describe a method to create optimal linear spline approximations to arbitrary functions of one or two variables, given as scattered data without known connectivity. We start with an initial approximation consisting of a fixed number of vertices and improve this approximation by choosing different ver-tices, governed by a simulated annealing algorithm. In the case of one variable, the approximation is defined by line segments; in the case of two variables, the vertices are connected to define a Delaunay triangulation of the selected subset of sites in the plane. In a second version of this algorithm, specifically designed for the bivariate case, we choose vertex sets and also change the triangulation to achieve both optimal vertex placement and optimal triangulation. We then cre-ate a hierarchy of linear spline approximations, each one being a superset of all lower-resolution ones.

On Curved Simplicial Elements and Best Quadratic Spline Approximation for Hierarchical Data Representation

We present a method for hierarchical data approximation using curved quadratic simplicial elements for domain decomposition. Scientific data defined over two- or three-dimensional domains typically contain boundaries and discontinuities that are to be preserved and approximated well for data analysis and visualization. Curved simplicial elements make possible a better representation of curved geometry, domain boundaries, and discontinuities than simplicial elements with non-curved edges and faces. We use quadratic basis functions and compute best quadratic simplicial spline approximations that are $C^0$-continuous everywhere except where field discontinuities occur whose locations we assume to be given. We adaptively refine a simplicial approximation by identifying and bisecting simplicial elements with largest errors. It is possible to store multiple approximation levels of increasing quality. Our method can be used for hierarchical data processing and visualization

SOME RESULTS ON THE DESIGN OF EXPERIMENTS FOR COMPARING UNREPLICATED TREATMENTS

In early generation variety trials, large numbers of new varieties may be compared, and little seed is usually available for each variety. A so-called unreplicated trial has each new variety on just one plot at a site, but includes several (often around 5) replicated check or control (or standard) varieties. The total proportion of check plots is usually between 10% and 20%. The aim of the trial is to choose some (around 1/3) good performing varieties to go on for further testing, rather than precise estimation of their mean yield. Now that spatial analyses of data from field experiments are becoming more common, there is interest in an efficient layout of an experiment given a proposed spatial analysis. Some possible design criteria are discussed, and efficient layouts under spatial dependence are considered

Metamodeling for the quantitative assessment of conceptual designs in an immersive virtual reality environment

The engineering design process has undergone extensive research in the area of detailed design. Many computer aided design (CAD) software packages have been developed from this research to provide an integral analysis tool for companies in the detailed design phase. However with the development of more complex technologies and systems, decisions made earlier in the design process have been crucial to product success. To help provide valuable information to assist these earlier decisions, tools have also been developed for conceptual design such as lightened CAD packages, concept elimination methods, and image processing software. Unfortunately, these tools have been proven ineffective based on the inability to provide a lower fidelity real-time analysis of each and every concept. By providing real-time analysis, engineers could spend more time evaluating every concept mathematically and base decisions on factual information instead of personal opinion. On a different note, companies continually undergo next generation development of their products. This continuous cycle of design iterations generates a stockpile of high fidelity analysis which we refer to as legacy data. Legacy data contains thousands of geometrical properties and analytical data used to assess the validity of previous designs. This data creates a vast amount of analytical engineering knowledge which can be harnessed to help evaluate the validity of future designs. Statistical approximations known as metamodels can be applied to summarize the general trends of the inputs and outputs of legacy dataset, and eliminate the need for recreating CAD analysis models for each concept. Metamodeling techniques cannot produce 100% accuracy, but at the conceptual design stage, 100% accuracy is not a necessity. This thesis presents an implementation scheme for incorporating Polynomial Response Surface (PRS) methods, Kriging Approximations, and Radial Basis Function Neural Networks (RBFNN) into conceptual design. A conceptual design software application, the Advanced Systems Design Suite (ASDS), has also been developed to incorporate these metamodeling techniques into assessment tools to evaluate conceptual design concepts in both a desktop and immersive virtual reality (VR) environment. The goal of the implementation scheme was to develop a strategy for constructing metamodels upon conceptual design datasets based upon their ability to perform under several conditions including various sample sizes, dataset linearity, interpolation within a domain, and extrapolation outside a domain. In order to develop the implementation scheme, two conceptual design datasets, wheel loading and stress analysis, were constructed due to a lack of available legacy data. The two datasets were setup using a design of experiments (DOE) to generate accurate sample points for the datasets. Once the DOE was formulated, digital prototypes were created in CAD software and the FEA test runs generated the responses of the DOE input parameters. The results of these FEA simulations generated the necessary conceptual design datasets required analyze the three metamodeling techniques. The performance results revealed that each metamodeling technique outperformed the others when tested again the various parameters. For instance, PRS metamodels performed very well when extrapolating outside its domain and with datasets consisting of more than 40 sample points. PRS metamodels require very setup and can be generated very quickly. If speed is the key consideration for metamodel construction, then PRS is the best option. Kriging metamodels showed the best performance with any non-linear dataset and large design space datasets exhibiting linear or non-linear behavior. Kriging metamodels are a very robust metamodeling technique especially when using a first-order global model on non-linear datasets. On the downside, Kriging metamodels require slightly more time to setup and construct than PRS metamodels. RBFNN metamodels performed well when interpolating within a large design space and on any sample size of linear datasets. However to reach performance levels of either PRS or Kriging, the ideal radius value must be determined prior to constructing the final model which took hours on small datasets. If the datasets consisted of thousands of design variables, constructing a RBFNN metamodel would take days to weeks to generate. However if construction time is not an issue, RBFNN metamodels outperform both PRS and Kriging techniques on linear datasets. This implementation scheme for incorporating metamodels into conceptual design provides a method for generating rapid assessment capabilities as an alternative to high fidelity analysis. Future work includes evaluating additional conceptual design datasets to create a more robust implementation scheme. More research will also be done in implementing additional types and varying setup parameters of both Kriging Approximations and Radial Basis Function Neural Networks

State-of-the-art in aerodynamic shape optimisation methods

Aerodynamic optimisation has become an indispensable component for any aerodynamic design over the past 60 years, with applications to aircraft, cars, trains, bridges, wind turbines, internal pipe flows, and cavities, among others, and is thus relevant in many facets of technology. With advancements in computational power, automated design optimisation procedures have become more competent, however, there is an ambiguity and bias throughout the literature with regards to relative performance of optimisation architectures and employed algorithms. This paper provides a well-balanced critical review of the dominant optimisation approaches that have been integrated with aerodynamic theory for the purpose of shape optimisation. A total of 229 papers, published in more than 120 journals and conference proceedings, have been classified into 6 different optimisation algorithm approaches. The material cited includes some of the most well-established authors and publications in the field of aerodynamic optimisation. This paper aims to eliminate bias toward certain algorithms by analysing the limitations, drawbacks, and the benefits of the most utilised optimisation approaches. This review provides comprehensive but straightforward insight for non-specialists and reference detailing the current state for specialist practitioners

Recovery of the Shear Modulus and Residual Stress of Hyperelastic Soft Tissues by Inverse Spectral Techniques

Inverse spectral techniques are developed in this dissertation for recovering the shear modulus and residual stress of soft tissues. Shear modulus is one of several quantities for measuring the stiffness of a material, and hence estimating it accurately is an important factor in tissue characterization. Residual stress is a stress that can exist in a body in the absence of externally applied loads, and beneficial for biological growth and remodeling. It is a challenge to recover the two quantities in soft tissues both theoretically and experimentally. The current inverse spectral techniques recover the two unknowns invasively, and are theoretically based on a novel use of the intravascular ultrasound technology (IVUS) by obtaining several natural frequencies of the vessel wall material. As the IVUS is interrogating inside the artery, it produces small amplitude, high frequency time harmonic vibrations superimposed on the quasistatic deformation of the blood pressure pre-stressed and residually stressed artery. The arterial wall is idealized as a nonlinear isotropic cylindrical hyperelastic body for computational convenience. A boundary value problem is formulated for the response of the arterial wall within a specific class of quasistatic deformations reflexive of the response due to imposed blood pressures. Subsequently, a boundary value problem is developed from intravascular ultrasound interrogation generating small amplitude, high frequency time harmonic vibrations superimposed on the quasistatic finite deformations via an asymptotic construction of the solutions. This leads to a system of second order ordinary Sturm-Liouville problems (SLP) with the natural eigenfrequencies from IVUS implementation as eigenvalues of the SLP. They are then employed to reconstruct the shear modulus and residual stress in a nonlinear approach by inverse spectral techniques. The shear modulus is recovered by a multidimensional secant method (MSM). The MSM avoids computing the Jacobian matrix of the equations and is shown to be convenient for manipulation. Residual stress is recovered via an optimization approach (OA) instead of the traditional equation-solving method. The OA increases the robustness of the algorithms by overdetermination of the problem, and comprehensive tests are performed to guarantee the accuracy of the solution. Numerical examples are displayed to show the viability of these techniques