Curve Skeleton and Moments of Area Supported Beam Parametrization in Multi-Objective Compliance Structural Optimization

This work addresses the end-to-end virtual automation of structural optimization up to the derivation of a parametric geometry model that can be used for application areas such as additive manufacturing or the verification of the structural optimization result with the finite element method. A holistic design in structural optimization can be achieved with the weighted sum method, which can be automatically parameterized with curve skeletonization and cross-section regression to virtually verify the result and control the local size for additive manufacturing. is investigated in general. In this paper, a holistic design is understood as a design that considers various compliances as an objective function. This parameterization uses the automated determination of beam parameters by so-called curve skeletonization with subsequent cross-section shape parameter estimation based on moments of area, especially for multi-objective optimized shapes. An essential contribution is the linking of the parameterization with the results of the structural optimization, e.g., to include properties such as boundary conditions, load conditions, sensitivities or even density variables in the curve skeleton parameterization. The parameterization focuses on guiding the skeletonization based on the information provided by the optimization and the finite element model. In addition, the cross-section detection considers circular, elliptical, and tensor product spline cross-sections that can be applied to various shape descriptors such as convolutional surfaces, subdivision surfaces, or constructive solid geometry. The shape parameters of these cross-sections are estimated using stiffness distributions, moments of area of 2D images, and convolutional neural networks with a tailored loss function to moments of area. Each final geometry is designed by extruding the cross-section along the appropriate curve segment of the beam and joining it to other beams by using only unification operations. The focus of multi-objective structural optimization considering 1D, 2D and 3D elements is on cases that can be modeled using equations by the Poisson equation and linear elasticity. This enables the development of designs in application areas such as thermal conduction, electrostatics, magnetostatics, potential flow, linear elasticity and diffusion, which can be optimized in combination or individually. Due to the simplicity of the cases defined by the Poisson equation, no experts are required, so that many conceptual designs can be generated and reconstructed by ordinary users with little effort. Specifically for 1D elements, a element stiffness matrices for tensor product spline cross-sections are derived, which can be used to optimize a variety of lattice structures and automatically convert them into free-form surfaces. For 2D elements, non-local trigonometric interpolation functions are used, which should significantly increase interpretability of the density distribution. To further improve the optimization, a parameter-free mesh deformation is embedded so that the compliances can be further reduced by locally shifting the node positions. Finally, the proposed end-to-end optimization and parameterization is applied to verify a linear elasto-static optimization result for and to satisfy local size constraint for the manufacturing with selective laser melting of a heat transfer optimization result for a heat sink of a CPU. For the elasto-static case, the parameterization is adjusted until a certain criterion (displacement) is satisfied, while for the heat transfer case, the manufacturing constraints are satisfied by automatically changing the local size with the proposed parameterization. This heat sink is then manufactured without manual adjustment and experimentally validated to limit the temperature of a CPU to a certain level.:TABLE OF CONTENT III I LIST OF ABBREVIATIONS V II LIST OF SYMBOLS V III LIST OF FIGURES XIII IV LIST OF TABLES XVIII 1. INTRODUCTION 1 1.1 RESEARCH DESIGN AND MOTIVATION 6 1.2 RESEARCH THESES AND CHAPTER OVERVIEW 9 2. PRELIMINARIES OF TOPOLOGY OPTIMIZATION 12 2.1 MATERIAL INTERPOLATION 16 2.2 TOPOLOGY OPTIMIZATION WITH PARAMETER-FREE SHAPE OPTIMIZATION 17 2.3 MULTI-OBJECTIVE TOPOLOGY OPTIMIZATION WITH THE WEIGHTED SUM METHOD 18 3. SIMULTANEOUS SIZE, TOPOLOGY AND PARAMETER-FREE SHAPE OPTIMIZATION OF WIREFRAMES WITH B-SPLINE CROSS-SECTIONS 21 3.1 FUNDAMENTALS IN WIREFRAME OPTIMIZATION 22 3.2 SIZE AND TOPOLOGY OPTIMIZATION WITH PERIODIC B-SPLINE CROSS-SECTIONS 27 3.3 PARAMETER-FREE SHAPE OPTIMIZATION EMBEDDED IN SIZE OPTIMIZATION 32 3.4 WEIGHTED SUM SIZE AND TOPOLOGY OPTIMIZATION 36 3.5 CROSS-SECTION COMPARISON 39 4. NON-LOCAL TRIGONOMETRIC INTERPOLATION IN TOPOLOGY OPTIMIZATION 41 4.1 FUNDAMENTALS IN MATERIAL INTERPOLATIONS 43 4.2 NON-LOCAL TRIGONOMETRIC SHAPE FUNCTIONS 45 4.3 NON-LOCAL PARAMETER-FREE SHAPE OPTIMIZATION WITH TRIGONOMETRIC SHAPE FUNCTIONS 49 4.4 NON-LOCAL AND PARAMETER-FREE MULTI-OBJECTIVE TOPOLOGY OPTIMIZATION 54 5. FUNDAMENTALS IN SKELETON GUIDED SHAPE PARAMETRIZATION IN TOPOLOGY OPTIMIZATION 58 5.1 SKELETONIZATION IN TOPOLOGY OPTIMIZATION 61 5.2 CROSS-SECTION RECOGNITION FOR IMAGES 66 5.3 SUBDIVISION SURFACES 67 5.4 CONVOLUTIONAL SURFACES WITH META BALL KERNEL 71 5.5 CONSTRUCTIVE SOLID GEOMETRY 73 6. CURVE SKELETON GUIDED BEAM PARAMETRIZATION OF TOPOLOGY OPTIMIZATION RESULTS 75 6.1 FUNDAMENTALS IN SKELETON SUPPORTED RECONSTRUCTION 76 6.2 SUBDIVISION SURFACE PARAMETRIZATION WITH PERIODIC B-SPLINE CROSS-SECTIONS 78 6.3 CURVE SKELETONIZATION TAILORED TO TOPOLOGY OPTIMIZATION WITH PRE-PROCESSING 82 6.4 SURFACE RECONSTRUCTION USING LOCAL STIFFNESS DISTRIBUTION 86 7. CROSS-SECTION SHAPE PARAMETRIZATION FOR PERIODIC B-SPLINES 96 7.1 PRELIMINARIES IN B-SPLINE CONTROL GRID ESTIMATION 97 7.2 CROSS-SECTION EXTRACTION OF 2D IMAGES 101 7.3 TENSOR SPLINE PARAMETRIZATION WITH MOMENTS OF AREA 105 7.4 B-SPLINE PARAMETRIZATION WITH MOMENTS OF AREA GUIDED CONVOLUTIONAL NEURAL NETWORK 110 8. FULLY AUTOMATED COMPLIANCE OPTIMIZATION AND CURVE-SKELETON PARAMETRIZATION FOR A CPU HEAT SINK WITH SIZE CONTROL FOR SLM 115 8.1 AUTOMATED 1D THERMAL COMPLIANCE MINIMIZATION, CONSTRAINED SURFACE RECONSTRUCTION AND ADDITIVE MANUFACTURING 118 8.2 AUTOMATED 2D THERMAL COMPLIANCE MINIMIZATION, CONSTRAINT SURFACE RECONSTRUCTION AND ADDITIVE MANUFACTURING 120 8.3 USING THE HEAT SINK PROTOTYPES COOLING A CPU 123 9. CONCLUSION 127 10. OUTLOOK 131 LITERATURE 133 APPENDIX 147 A PREVIOUS STUDIES 147 B CROSS-SECTION PROPERTIES 149 C CASE STUDIES FOR THE CROSS-SECTION PARAMETRIZATION 155 D EXPERIMENTAL SETUP 15

Verification and Validation of the Spalart-Allmaras Turbulence Model for Strand Grids

The strand-Cartesian grid approach is a unique method of generating and computing fluid dynamic simulations. The strand-Cartesian approach provides highly desirable qualities of fully-automatic grid generation and high-order accuracy. This thesis focuses on the implementation of the Spalart-Allmaras turbulence model to the strand-Cartesian grid framework. Verification and validation is required to ensure correct implementation of the turbulence model.Mathematical code verification is used to ensure correct implementation of new algorithms within the code framework. The Spalart-Allmaras model is verified with the Method of Manufactured Solutions (MMS). MMS shows second-order convergence, which implies that the new algorithms are correctly implemented.Validation of the strand-Cartesian solver is completed by simulating certain cases for comparison against the results of two independent compressible codes; CFL3D and FUN3D. The NASA-Langley turbulence resource provided the inputs and conditions required to run the cases, as well as the case results for these two codes. The strand solver showed excellent agreement with both NASA resource codes for a zero-pressure gradient flat plate and bump- in-channel. The treatment of the sharp corner on a NACA 0012 airfoil is investigated, resulting in an optimal external sharp corner configuration of strand vector smoothing with a base Cartesian grid and telescoping Cartesian refinement around the trailing edge. Results from the case agree well with those from CFL3D and FUN3D. Additionally, a NACA 4412 airfoil case is examined, and shows good agreement with CFL3D and FUN3D, resulting in validation for this case

Structure Preserving and Scalable Simulation of Colliding Systems

Predictive computational tools to study granular materials are important in fields ranging from the geosciences and civil engineering to computer graphics. The simulation of granular materials, however, presents many challenges. The behavior of a granular medium is fundamentally multi-scale, with pair-wise interactions between discrete granules able to influence the continuum-scale evolution of a bulk material. Computational techniques for studying granular materials must therefore contend with this multi-scale nature. This research first addresses both the question of how to accurately model interactions between grains and the question of how to achieve multi-scale simulations of granular materials. We propose a novel rigid body contact model and a time integration technique that, for the first time, are able to simultaneously capture five key features of rigid body impact. We further validate this new model and time integration method by reproducing computationally challenging phenomena from granular physics. We next propose a technique to couple discrete and continuum models of granular materials to one another. This hybrid model reveals a family of possible discretizations suitable for simulation. We derive an explicit integration technique from this framework that is able to capture phenomena previously reserved for discrete treatments, including frictional jamming, while treating bulk regions of the material with a continuum model. To effectively handle the large plastic deformations inherent in the evolution of a granular medium, we further propose a method to dynamically update which regions are treated with a discrete model and which regions are treated with a continuum model. We demonstrate that hybrid simulations of a dynamically evolving granular material are possible and practical, and lay the foundation for further algorithmic development in this space. Finally, as the the tools used in computational science and engineering become progressively more complex, the ability to effectively train students in the field becomes increasingly important. We address the question of how to train students from a computer science background in numerical computation techniques by proposing a new system to automatically vet and identify problems in numerical simulations. This system has been deployed at the undergraduate and graduate level in a course on physical simulation at Columbia University, and has increased both student retention and student satisfaction with the course

Efficient Large Scale Transient Heat Conduction Analysis Using A Parallelized Boundary Element Method

A parallel domain decomposition Laplace transform Boundary Element Method, BEM, algorithm for the solution of large-scale transient heat conduction problems will be developed. This is accomplished by building on previous work by the author and including several new additions (most note-worthy is the extension to 3-D) aimed at extending the scope and improving the efficiency of this technique for large-scale problems. A Laplace transform method is utilized to avoid time marching and a Proper Orthogonal Decomposition, POD, interpolation scheme is used to improve the efficiency of the numerical Laplace inversion process. A detailed analysis of the Stehfest Transform (numerical Laplace inversion) is performed to help optimize the procedure for heat transfer problems. A domain decomposition process is described in detail and is used to significantly reduce the size of any single problem for the BEM, which greatly reduces the storage and computational burden of the BEM. The procedure is readily implemented in parallel and renders the BEM applicable to large-scale transient conduction problems on even modest computational platforms. A major benefit of the Laplace space approach described herein, is that it readily allows adaptation and integration of traditional BEM codes, as the resulting governing equations are time independent. This work includes the adaptation of two such traditional BEM codes for steady-state heat conduction, in both two and three dimensions. Verification and validation example problems are presented which show the accuracy and efficiency of the techniques. Additionally, comparisons to commercial Finite Volume Method results are shown to further prove the effectiveness

Research reports: 1991 NASA/ASEE Summer Faculty Fellowship Program

The basic objectives of the programs, which are in the 28th year of operation nationally, are: (1) to further the professional knowledge of qualified engineering and science faculty members; (2) to stimulate an exchange of ideas between participants and NASA; (3) to enrich and refresh the research and teaching activities of the participants' institutions; and (4) to contribute to the research objectives of the NASA Centers. The faculty fellows spent 10 weeks at MSFC engaged in a research project compatible with their interests and background and worked in collaboration with a NASA/MSFC colleague. This is a compilation of their research reports for summer 1991

High-Performance Integrated Window and Façade Solutions for California

The researchers developed a new generation of high-performance façade systems and supporting design and management tools to support industry in meeting California’s greenhouse gas reduction targets, reduce energy consumption, and enable an adaptable response to minimize real-time demands on the electricity grid. The project resulted in five outcomes: (1) The research team developed an R-5, 1-inch thick, triplepane, insulating glass unit with a novel low-conductance aluminum frame. This technology can help significantly reduce residential cooling and heating loads, particularly during the evening. (2) The team developed a prototype of a windowintegrated local ventilation and energy recovery device that provides clean, dry fresh air through the façade with minimal energy requirements. (3) A daylight-redirecting louver system was prototyped to redirect sunlight 15–40 feet from the window. Simulations estimated that lighting energy use could be reduced by 35–54 percent without glare. (4) A control system incorporating physics-based equations and a mathematical solver was prototyped and field tested to demonstrate feasibility. Simulations estimated that total electricity costs could be reduced by 9-28 percent on sunny summer days through adaptive control of operable shading and daylighting components and the thermostat compared to state-of-the-art automatic façade controls in commercial building perimeter zones. (5) Supporting models and tools needed by industry for technology R&D and market transformation activities were validated. Attaining California’s clean energy goals require making a fundamental shift from today’s ad-hoc assemblages of static components to turnkey, intelligent, responsive, integrated building façade systems. These systems offered significant reductions in energy use, peak demand, and operating cost in California

A verification of steady state discontinuous solutions using the method of manufactured solutions for finite volume computational fluid dynamic codes

When applying the method of manufactured solutions (MMS) on computational fluid dynamic (CFD) software, it is traditionally a requirement that all solutions be continuous on the computational domain. This stipulation is limiting for the verification and validation of CFD solutions where discontinuities are frequent. This work details the development of a discontinuous MMS method for finite volume codes. The CFD code used throughout this research is a cell centered, finite volume, 1st order, Eulerian scheme within the software AVUS (Air Vehicles Unstructured Solver) which is combined with uniform structured grids. This code is used as a representative testing platform with the convenience of accessible source code. A piecewise technique is used for defining manufactured solutions which simulate discontinuities. Since source terms which allow arbitrary solutions in continuous MMS do not exist within Riemann solvers, conditions at the shock boundary are physically constrained by the Rankine-Hugoniot jump conditions. Upwind manufactured solutions are first initialized and a regression technique is then used to solve for solutions downwind of the discontinuity. It is shown that a change in regression error of four order of magnitude has no significant effect on an order of convergence test. When applying MMS on finite volume CFD codes, determining the exact solutions and source terms when the stored value is the integrated average over the control volume is a non-trivial and frequently ignored problem. MMS with discontinuities further complicates the problem of determining these values. To obtain low error and high convergence rates, linearly and quadratically exact transformations are proposed for cells split by discontinuities. These transformations are combined with a nine point Gauss quadrature method to achieve 4th order accuracy for fully general solutions and shock shapes. To begin testing, continuous MMS is first performed to ensure a verified code. AVUS is verified for 1st order solutions but retains lower order boundary conditions when solving 2nd order. The error is verified using a second academic CFD solver but is left unchanged for shock solutions which are inherently 1st order. Constant primitive, oblique shock solutions are then used to demonstrate a solution\u27s error dependence on grid alignment. Grid alignment is shown to play a vital role in the error surrounding a shock. Constant oblique solutions with a grid aligned shock result in no discretization error while a shock that passed through the interior of cells yields error upwards of 4% for the u-component velocity. A semi one-dimensional problem combined with a grid aligned shock is then used to demonstrate the error magnitude (\u3c 1%) due to the cell averaging on both sides of the discontinuity. Fully generic primitives and discontinuities are then introduced and grid convergence studies yielding 1st order results typically associated with shocks are used to verify the correctness of the code. Despite high errors near the region of the shock, similar patterns and orders of convergence are shown for both physical and mathematical shock shapes. Sub-linear convergences near 0.9, especially in the u-component velocity, indicate the presence of linearly degenerate characteristics which are typical of shock capturing schemes. The fully general solutions are used to show that errors in the Riemann solver can be identified with discontinuous MMS. Three coding errors are introduced which are not identified by continuous MMS. In two cases the discontinuous procedure indicates a coding error in the convergence test and the final error is classified as a efficiency mistake and is missed by all methods. Lastly, the method is replicated on an academic CFD code for a final validation procedure. Identical behavior and near identical errors suggest the robustness of the developed method despite fundamental differences in the shock capturing schemes of the two codes

Lidar In Coastal Storm Surge Modeling: Modeling Linear Raised Features

A method for extracting linear raised features from laser scanned altimetry (LiDAR) datasets is presented. The objective is to automate the method so that elements in a coastal storm surge simulation finite element mesh might have their edges aligned along vertical terrain features. Terrain features of interest are those that are high and long enough to form a hydrodynamic impediment while being narrow enough that the features might be straddled and not modeled if element edges are not purposely aligned. These features are commonly raised roadbeds but may occur due to other manmade alterations to the terrain or natural terrain. The implementation uses the TauDEM watershed delineation software included in the MapWindow open source Geographic Information System to initially extract watershed boundaries. The watershed boundaries are then examined computationally to determine which sections warrant inclusion in the storm surge mesh. Introductory work towards applying image analysis techniques as an alternate means of vertical feature extraction is presented as well. Vertical feature lines extracted from a LiDAR dataset for Manatee County, Florida are included in a limited storm surge finite element mesh for the county and Tampa Bay. Storm surge simulations using the ADCIRC-2DDI model with two meshes, one which includes linear raised features as element edges and one which does not, verify the usefulness of the method