Data-Driven M-Vcut Topology Optimization Method for Heat Conduction Problem of Cellular Structure with Multiple Microstructure Prototypes

High-order three-scale computational method for heat conduction problems of axisymmetric composite structures with multiple spatial scales

Multiscale computational method for heat conduction problems of composite structures with diverse periodic configurations in different subdomains

Heat conduction [1] is usually modeled as a diffusion process embodied in heat conduction equation. The traditional numerical methods [2, 3] for heat conduction problems such as the finite difference or finite element are well developed. However, these methods are based on discretized mesh systems, thus they are inherently limited in the geometry treatment. This chapter describes the Monte Carlo method that is based on particle transport simulation to solve heat conduction problems. The Monte Carlo method is “meshless” and thus can treat problems with very complicated geometries. The method is applied to a pebble fuel to be used in very high temperature gas-cooled reactors (VHTGRs) [4], which is a next-generation nuclear reactor being developed. Typically, a single pebble contains ~10,000 particle fuels randomly dispersed in graphite– matrix. Each particle fuel is in turn comprised of a fuel kernel and four layers of coatings. Furthermore, a typical reactor would house several tens of thousands of pebbles in the core depending on the power rating of the reactor. See Fig. 1. Such a level of geometric complexity and material heterogeneity defies the conventional mesh–based computational methods for heat conduction analysis. Among transport methods, the Monte Carlo method, that is based on stochastic particle simulation, is widely used in neutron and radiation particle transport problems such as nuclear reactor design. The Monte Carlo method described in this chapter is based on the observation that heat conduction is a diffusion process whose governing equation is analogous to the neutron diffusion equation [5] under no absorption, no fission and one speed condition, which is a special form of the particle transport equation. While neutron diffusion approximates the neutron transport phenomena, conversely it is applicable to solve diffusion problems by transport methods under certain conditions. Based on this idea, a new Monte Carlo method has been recently developed [6-8] to solve heat conduction problems. The method employs the MCNP code [9] as a major computational engine. MCNP is a widely used Monte Carlo particle transport code with versatile geometrical capabilities. Monte Carlo techniques for heat conduction have been reported [10-13] in the past. But most of the earlier Monte Carlo methods for heat conduction are based on discretized mesh systems, thus they are limited in the capabilities of geometry treatment. Fraley et al[13] uses a “meshless” system like the method in this chapter but does not give proper treatment to the boundary conditions, nor considers the “diffusion-transport theory correspondence” to be described in Section 2.2 in this chapter. Thus, the method in this chapter is a transport theory treatment of the heat conduction equation with a methodical boundary correction. The

A singular integral method of numerical analysis for two-dimensional steady-state heat conduction problems with any combination of temperature, gradient, or convection boundary conditions is presented. Excellent agreement with the exact solution is illustrated for an example problem. The method is used to determine the solution for a fin bank with convection.

Numerical experiments using hierarchical finite element method for nonlinear heat conduction in plates

The fully ceramic microencapsulated (FCM) fuel, one of the accident tolerant fuel (ATF) concepts, consists of TRISO particles randomly dispersed in SiC matrix. This high heterogeneity in compositions leads to difficulty in explicit thermal calculation of such a fuel. For thermal analysis of a fuel element in very high temperature reactors (VHTRs) which has a similar configuration to the FCM fuel, a two-temperature homogenized model was proposed by the authors. The model was developed using particle transport Monte Carlo method for heat conduction problems. It gives more realistic temperature profiles and provides the fuel-kernel and graphite temperatures separately. In this paper, we apply the two-temperature homogenized model to single-channel thermal analysis of the FCM fuel element for steady- and transient-states using 2-D FEM/1-D FDM hybrid method. The results of analyses are compared to those of conventional UO2 fuel having the same geometric dimension and operating conditions.

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This paper presents an efficient and comprehensive MATLAB code to solve two-dimensional structural topology optimization problems, including minimum mean compliance, compliant mechanism synthesis and multi-load compliance problems. The Unsmooth Variational Topology Optimization (UNVARTOP) method, developed by the authors in a previous work, is used in the topology optimization code, based on the finite element method (FEM), to compute the sensitivity and update the topology. The paper also includes instructions to improve the bisection algorithm, modify the computation of the Lagrangian multiplier by using an Augmented Lagrangian to impose the constraint, implement heat conduction problems and extend the code to three-dimensional topology optimization problems. The code, intended for students and newcomers in topology optimization, is included as an appendix (Appendix A) and it can be downloaded from https://github.com/DanielYago together with supplementary material.

To date the design of structures via topology optimisation methods has mainly focused on single-objective problems. However, real-world design problems usually involve several different objectives, most of which counteract each other. Therefore, designers typically seek a set of Pareto optimal solutions, a solution for which no other solution is better in all objectives, which capture the trade-off between these objectives. This set is known as a smart Pareto set. Currently, only the weighted sums method has been used for generating Pareto fronts with topology optimisation methods. However, the weighted sums method is unable to produce evenly distributed smart Pareto sets. Furthermore, evenly distributed weights have been shown to not produce evenly spaced solutions. Therefore, the weighted sums method is not suitable for generating smart Pareto sets. Recently, the smart normal constraints method has been shown to be capable of directly generating smart Pareto sets. This work presents an updated smart normal constraint method, which is combined with a bi-directional evolutionary structural optimisation algorithm for multi-objective topology optimisation. The smart normal constraints method has been modified by further restricting the feasible design space for each optimisation run such that dominant and redundant points are not found. The algorithm is tested on several different structural optimisation problems. A number of different structural objectives are analysed, namely compliance, dynamic and buckling objectives. Therefore, the method is shown to be capable of solving various types of multi-objective structural optimisation problems. The goal of this work is to show that smart Pareto sets can be produced for complex topology optimisation problems. Furthermore, this research hopes to highlight the gap in the literature of topology optimisation for multi-objective problems.

Data-driven computing for nonlinear problems of composite structures based on sub-domain search technique

This paper deals with simultaneous topology and shape optimization of elastic contact problems. The structural optimization problem for an elastic contact problem is formulated. Shape as well as topological derivatives formulae of the cost functional are provided using material derivative and asymptotic expansion methods, respectively. These derivatives are employed to formulate necessary optimality condition for simultaneous shape and topology optimization and to calculate a descent direction in numerical algorithm. Level set based numerical algorithm for the solution of this optimization problem is proposed. Numerical examples are provided and discussed.

A continuous variable optimization method and a topological optimization method are proposed for the vibration control of piezoelectric truss structures by means of the optimal placements of active bars. In this optimization model, a zero-one discrete variable is defined in order to solve the optimal placement of piezoelectric active bars. At the same time, the feedback gains are also optimized as continuous design variables. A two-phase procedure is proposed to solve the optimization problem. The sequential linear programming algorithm is used to solve optimization problem and the sensitivity analysis is carried out for objective and constraint functions to make linear approximations. On the basis of the Newmark time integration of structural transient dynamic responses, a new sensitivity analysis method is developed in this paper for the vibration control problem of piezoelectric truss structures with respect to various kinds of design variables. Numerical examples are given in the paper to demonstrate the effectiveness of the methods.

A globally convergent method to accelerate topology optimization using on-the-fly model reduction

The center of mass and the inertia tensor are important parameters characterizing the kinetic properties of many structural systems, e.g., the rotor, spacecraft, and aircraft. In these occasions, designs not consistent with the requirements on the center of mass and inertia tensor cannot be accepted. In this paper, mathematical models are established to carry out the topology optimization of continua considering the mass and inertia characteristics. Two numerical examples, one for minimizing the static compliance considering locations of the center of mass and the other for maximizing the fundamental eigenfrequency considering both locations of the center of mass and values of the inertia products, are presented to demonstrate that the requirements on mass and inertia characteristics could have a profound influence on the final optimized layouts in structural topology optimization problems. On the other hand, the mass and inertia characteristics can also be tuned by topology optimization methods. In the second numerical problem, the smooth minimum operator is used to approximate the fundamental eigenfrequency with the first 4 eigenfrequencies so that the possible non-differentiability of the repeated eigenfrequencies can be avoided. Due to its extreme importance in engineering practice, the work of this paper can be useful to carry out designs in accordance with the requirements on the mass and inertia characteristics under the framework of traditional density-based structural topology optimization methods.

The numerical method of stamp topological optimization taking into account fatigue strength is presented in the work. It is proposed to take into account the restrictions on the stress state in accordance with the curve of the dependence of the maximum stresses on the number of loading cycles in the ESO topological optimization method. An approach to the selection of the evolutionary coefficient with a step-by-step increase in the rejection coefficient is proposed when constructing an iterative scheme for the rejection of elements by the method of topological optimization. The calculation of the stamp optimal topology with a decrease in volume due to the removal and redistribution of material was carried out in the study. The new geometric model of the optimal topology stamp is based on the predicted distribution of elements with a minimum stress level. The verification calculation of the stress state of the stamp of optimal topology with an assessment of fatigue strength was carried out in the work. The numerical calculation was carried out using the finite element method in the Ansys software package. The minimized stamp volume decreased by 35% according to the calculation results. The results of the study can be further applied in the development of topological optimization methods and in the design of stamping tools of optimal topology.