12,464 research outputs found
Critical Transitions In a Model of a Genetic Regulatory System
We consider a model for substrate-depletion oscillations in genetic systems,
based on a stochastic differential equation with a slowly evolving external
signal. We show the existence of critical transitions in the system. We apply
two methods to numerically test the synthetic time series generated by the
system for early indicators of critical transitions: a detrended fluctuation
analysis method, and a novel method based on topological data analysis
(persistence diagrams).Comment: 19 pages, 8 figure
Nonlinear stability of stationary solutions for curvature flow with triple junction
In this paper we analyze the motion of a network of three planar curves with
a speed proportional to the curvature of the arcs, having perpendicular
intersections with the outer boundary and a common intersection at a triple
junction. As a main result we show that a linear stability criterion due to
Ikota and Yanagida is also sufficient for nonlinear stability. We also prove
local and global existence of classical smooth solutions as well as various
energy estimates. Finally, we prove exponential stabilization of an evolving
network starting from the vicinity of a linearly stable stationary network.Comment: submitte
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
Invariant Manifolds and Rate Constants in Driven Chemical Reactions
Reaction rates of chemical reactions under nonequilibrium conditions can be
determined through the construction of the normally hyperbolic invariant
manifold (NHIM) [and moving dividing surface (DS)] associated with the
transition state trajectory. Here, we extend our recent methods by constructing
points on the NHIM accurately even for multidimensional cases. We also advance
the implementation of machine learning approaches to construct smooth versions
of the NHIM from a known high-accuracy set of its points. That is, we expand on
our earlier use of neural nets, and introduce the use of Gaussian process
regression for the determination of the NHIM. Finally, we compare and contrast
all of these methods for a challenging two-dimensional model barrier case so as
to illustrate their accuracy and general applicability.Comment: 28 pages, 13 figures, table of contents figur
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