Fast Radial Basis Functions for Engineering Applications

✍ Scribed by Marco Evangelos Biancolini

• Publisher: Springer
• Year: 2018
• Tongue: English
• Leaves: 364
• Edition: 1st ed. 2017
• Category: Library

✦ Synopsis

This book presents the first “How To” guide to the use of radial basis functions (RBF). It provides a clear vision of their potential, an overview of ready-for-use computational tools and precise guidelines to implement new engineering applications of RBF.

Radial basis functions (RBF) are a mathematical tool mature enough for useful engineering applications. Their mathematical foundation is well established and the tool has proven to be effective in many fields, as the mathematical framework can be adapted in several ways. A candidate application can be faced considering the features of RBF:

multidimensional space (including 2D and 3D), numerous radial functions available, global and compact support, interpolation/regression.

This great flexibility makes RBF attractive – and their great potential has only been partially discovered. This is because of the difficulty in taking a first step toward RBF as they are not commonly part of engineers’ cultural background, but also due to the numerical complexity of RBF problems that scales up very quickly with the number of RBF centers. Fast RBF algorithms are available to alleviate this and high-performance computing (HPC) can provide further aid. Nevertheless, a consolidated tradition in using RBF in engineering applications is still missing and the beginner can be confused by the literature, which in many cases is presented with language and symbolisms familiar to mathematicians but which can be cryptic for engineers.

The book is divided in two main sections. The first covers the foundations of RBF, the tools available for their quick implementation and guidelines for facing new challenges; the second part is a collection of practical RBF applications in engineering, covering several topics, including response surface interpolation in n-dimensional spaces, mapping of magnetic loads, mapping of pressure loads, up-scaling of flow fields, stress/strain analysis by experimental displacement fields, implicit surfaces, mesh to cad deformation, mesh morphing for crack propagation in 3D, ice and snow accretion using computational fluid dynamics (CFD) data, shape optimization for external aerodynamics, and use of adjoint data for surface sculpting. For each application, the complete path is clearly and consistently exposed using the systematic approach defined in the first section.

✦ Table of Contents

Preface
Contents
Abstract
1 Introduction
Abstract
Reference
--- Fast RBF for Engineering
2 Radial Basis Functions
Abstract
2.1 RBF Theory Background
2.2 Radial Functions
2.3 A First Example of RBF at Work
2.3.1 Definition of the Source Points
2.3.2 Fitting the RBF
2.3.3 Evaluating the RBF
2.3.4 Exploring the Interpolated Function
2.3.5 Exploring Different Arrangements of Control Points
2.4 Differentiation of RBF
2.4.1 First Derivative
2.4.2 Example of First Derivatives Exploitation
2.4.3 Higher Order Derivatives
2.5 Fitting an RBF at Arbitrary Locations of the Known Data
2.5.1 A Simple Example
2.6 Interpolation Versus Regression
2.6.1 A Simple Example of RBF Regression
2.7 Approximation of Noisy Data
2.7.1 Noisy Data Fitting with a Relaxation Function
2.7.2 Noisy Data Fitting with Regression
References
3 Fast RBF
Abstract
3.1 Fast RBF Foundation
3.2 Direct Methods with Fast Libraries for Linear Algebra
3.3 Reduction of the Cloud Size (Greedy Methods)
3.3.1 A Simple Mesh Morphing Problem
3.4 Use of Compact Supported RBF
3.5 Partition of Unity Methods (POU)
3.5.1 Example of POU for Surface Projection
3.6 Fast Multipole Methods (FMM)
3.6.1 Multipole Expansion of the Bi-Harmonic Kernel
3.6.2 Approximated Expansion Using an RBF Function
3.6.3 Scaling Test Example Using Hierarchical FMM
3.7 Iterative Solvers
3.8 Parallel Solvers
3.8.1 Example of Acceleration Achieved with Parallel Solving Technology
References
4 RBF Tools
Abstract
4.1 RBF Fitting and Evaluating Functions
4.2 Aligned Points Check
4.3 Reducing the Size of the RBF Cloud (Sub-sampling)
4.4 Duplicates Check
4.5 Data Normalisation
4.6 Generation of Auxiliary RBF Clouds
4.6.1 Box
4.6.2 Cylinder
4.6.3 Sphere
References
--- Engineering Applications Examples
5 RBF Implicit Representation of Geometrical Entities
Abstract
5.1 Point Projection onto the RBF Implicit Surface
5.1.1 Processing of Target Mesh for Offset Points Generation
5.2 Classic Approach
5.3 New Method: Projection Field
5.4 Implicit Surface Representation Test Cases
5.4.1 Spherical shell
5.4.2 Add a Fillet on  a Cube Edge
References
6 RBF Mesh Morphing
Abstract
6.1 RBF Mesh Morphing Basic Principles
6.1.1 RBF Mesh Morphing Main Advantages and Drawbacks
6.1.2 RBF Mesh Morphing Application Scheme
6.1.3 RBF Mesh Morphing Strategy and RBF Solution Arrangement
6.2 Volume Mesh Morphing Paradigms
6.2.1 Delimiting Encapsulation Domains
6.2.2 Moving Encapsulation Domains
6.3 Surface Mesh Morphing Paradigms
6.3.1 Translation Modifier
6.3.2 Rotation Modifier
6.3.3 Scaling Modifier
6.3.4 Surface Offset Modifier
6.3.5 Projection Modifier
6.3.6 Free Form Deformation Using RBF
6.4 Multi-step Paradigm (Hierarchical Approach)
6.5 Multi-physics Data Driven Paradigm
6.6 CAD Driven Morphing
References
7 Advanced RBF Mesh Morphing for Biomechanical Applications
Abstract
7.1 Hemodynamics
7.1.1 Patient Specific Image-Based Hemodynamics: Parametric Study of the Carotid Bifurcation
7.1.2 A Virtual Test Bench for Hemodynamic Evaluation of Aortic Cannulation in Cardiopulmonary Bypass
7.2 Human Bones Stress Analysis
7.2.1 Tibia
7.2.2 Femur
References
8 Adjoint Sensitivities and RBF Mesh Morphing
Abstract
8.1 Adjoint Sensitivity Background
8.2 Role of Fast RBF in Adjoint Based Design
8.3 Adjoint Sculpting
8.3.1 RBF Morphing Set-Up for Adjoint Sculpting
8.3.2 Adjoint Sculpting Workflow
8.3.3 Filtering Tool for Sensitivity Data Processing
8.4 Adjoint Preview
8.4.1 Adjoint Preview Workflow
8.5 Structural Applications
8.5.1 Bracket
8.5.2 T-Beam
8.6 Fluid-Dynamic Applications
8.6.1 Taurus Glider
8.6.1.1 Glider Case: Adjoint Preview
8.6.1.2 Glider Case: Adjoint Sculpting
8.6.2 DrivAer Car Optimization
8.6.3 Airbox Runners Flow Balance Optimization
8.6.4 90° Curved Duct
References
9 Advanced RBF Mesh Morphing for Multi-Physics Applications with Evolutionary Shapes
Abstract
9.1 Mesh Morphing of Evolutionary Shapes
9.2 Snow Accretion
9.2.1 Computation of Snow Growth Using CFD
9.2.2 Pre Processing of Snow Data Using Mathcad
9.2.3 Exploring Accretion Data Using RBF
9.2.4 Mesh Update in the CFD Code
9.2.5 Validation of Snow Growth Predictions
9.3 Icing Simulation
9.4 Icing Simulation Tests Cases According to the Constrained Approach
9.4.1 2D Icing Case: NACA0012 and GLC305 Aerofoils
9.4.2 3D Icing Case: HIRENASD
9.5 Icing Simulation Tests Cases According to the On-The-Fly Approach
9.5.1 Adopted Strategy for the On-The-Fly Icing Implementation
9.5.2 On-The-Fly Icing Test Case CFD-Icing Tool
9.5.3 On-The-Fly Icing Verification
9.6 Crack Propagation
9.6.1 Semi-elliptical Crack on a Flat Bar
9.6.2 Edge Crack onto a Notched Round Bar
9.7 Biological Growth Method
9.7.1 Stress Reduction on a Cantilever Beam
9.7.2 Stress Reduction at a Turbine Blade Fillet
References
10 FSI Workflow using Advanced RBF Mesh Morphing
Abstract
10.1 Importance of FSI in Technical Applications
10.2 FSI Simulation Through RBF Mesh Morphing
10.3 Two-Way FSI Through RBF Mesh Morphing
10.3.1 Two-Way FSI Theoretical Background and Characterization
10.3.2 Two-Way FSI Through RBF Mesh Morphing
10.4 Mode-Superposition FSI Through RBF Mesh Morphing
10.4.1 Mode-Superposition FSI Theoretical Background
10.4.2 Mode-Superposition FSI Through RBF Mesh Morphing
10.5 Two-Way FSI Test Case
10.5.1 P1xx Wing
10.6 Mode-Superposition for Steady
10.6.1 P1xx Wing
10.6.2 HIRENASD Wing
10.6.3 Dallara INDY Race Car
10.7 Mode-Superposition for Transient FSI Analyses
10.7.1 Vortex Shedding Induced Vibrations
10.7.2 Store Separation
References
11 Optimization Workflows assisted by RBF Surrogate Models
Abstract
11.1 Optimisation Workflow
11.2 RBF Metamodeling
11.3 NACA 0012 Optimisation
11.3.1 Baseline Case Set-Up and Results
11.3.2 Optimization Assumptions and RBF Solutions Set-Up
11.3.3 Optimisation Results
11.4 Glider Optimisation
11.4.1 Baseline Case Set-Up and Results
11.4.2 Optimisation Assumptions and RBF Solution Set-Up
11.4.3 Optimisation Results
11.4.4 Optimal Candidate Selection and Optimal CAD Generation
11.5 Sails Trim Optimisation
11.5.1 Baseline CFD Case Set-Up
11.5.2 RBF Set-Up
11.5.3 Optimisation Results
References
12 Advanced Field Data Post-processing using RBF Interpolation
Abstract
12.1 Introduction to the Field Data Interpolation Problem
12.2 Evaluation of Equivalent Stresses According to the Implicit Gradient
12.2.1 Interpolation of the Stresses as Computed by FEA Solver
12.2.2 Interpolation of the Stresses Derived by FEA Displacements
12.3 Upscaling of FEA Results at Stress Raisers
12.3.1 Verification Procedure Steps and Mathematical Background of the Proposed Technique
12.3.2 Rectangular Plate with Single Centred Circular Hole
12.3.3 Plate with a Set of Centred Aligned Holes
12.3.4 Tube with a Single Transversal Hole
12.4 Isostatic Lines Construction for Planar Problems
12.5 Post-processing of Images Analysis of Structural Deformations
12.6 Compensation of Metrological Data
12.7 RBF for the Interpolation of Hemodynamics Flow Pattern
References
13 Data Mapping using RBF
Abstract
13.1 Data Mapping Background
13.2 Mapping of Pressure in Two-Way FSI Analyses
13.2.1 Introduction and Review of Mapping Schemes
13.2.2 Standard and High Fidelity RBF Mapping
13.2.3 Catenoid
13.2.4 DLR-F6
13.2.5 HIRENASD
13.3 Mapping of Electro-magnetic Loads in FEM Analyses
13.3.1 Introduction of the DEMO Project Context
13.3.2 Overview of the DEMO TF Coil System Design Scenario
13.3.3 DEMO TF Coil System FEM Analyses Through the Mapping Procedure Based on RBF
13.3.4 DEMO TF Coil System Analyses Results
13.3.5 Local Validation of Interpolated Magnetic Field
13.3.6 Resultant Loads Acting Along the TF Coil
13.3.7 Resultant Loads Acting on the TF Coil System WP
13.3.8 Stress Analysis of the Complete System
References