F. Navarrina

Block aggregation of stress constraints in topology optimization of structures

Structural topology optimization problems have been traditionally stated and solved by means of maximum stiffness formulations. On the other hand, some effort has been devoted to stating and solving this kind of problems by means of minimum weight formulations with stress (and/or displacement) constraints. It seems clear that the latter approach is closer to the engineering point of view, but it also leads to more complicated optimization problems, since a large number of highly non-linear (local) constraints must be taken into account to limit the maximum stress (and/or displacement) at the element level. In this paper, we explore the feasibility of defining a so-called global constraint, which basic aim is to limit the maximum stress (and/or displacement) simultaneously within all the structure by means of one single inequality. Should this global constraint perform adequately, the complexity of the underlying mathematical programming problem would be drastically reduced. However, a certain weakening of the feasibility conditions is expected to occur when a large number of local constraints are lumped into one single inequality. With the aim of mitigating this undesirable collateral effect, we group the elements into blocks. Then, the local constraints corresponding to all the elements within each block can be combined to produce a single aggregated constraint per block. Finally, we compare the performance of these three approaches (local, global and block aggregated constraints) by solving several topology optimization problems.

A boundary element numerical approach for grounding grid computation

Abstract Analysis and design of substation earthing involves computing the equivalent resistance of grounding systems, as well as distribution of potentials on the earth surface due to fault currents [1,2]. While very crude approximations were available in the 1960s, several methods have been proposed in the last two decades, most of them on the basis of intuitive ideas such as superposition of punctual current sources and error averaging [3,4]. Although these techniques represented a significant improvement in the area of earthing analysis, a number of problems have been reported; namely: large computational requirements, unrealistic results when segmentation of conductors is increased, and uncertainty in the margin of error [4]. A Boundary Element approach for the numerical computation of substation grounding systems is presented in this paper. Several widespread intuitive methods (such as the Average Potential Method) can be identified in this general formulation as the result of suitable assumptions introduced in the BEM formulation to reduce computational cost for specific choices of the test and trial functions. On the other hand, this general approach allows the use of linear and parabolic leakage current elements to increase accuracy. Efforts have been made particularly in getting a drastic reduction in computing time by means of completely new analytical integration techniques, while semi-iterative methods have proven to be specially efficient for solving the involved system of linear equations. This BEM formulation has been implemented in a specific Computer Aided Design system for grounding analysis developed within the last years. The feasibility of this approach is finally demonstrated by means of its application to two real problems.

Analysis of transferred Earth potentials in grounding systems: a BEM numerical approach

In this work, a numerical formulation is presented for the analysis of a common problem in electrical engineering practice, that is, the existence of transferred earth potentials in a grounding installation for metallic structures or conductors connected or not connected to the grounding grid . The transfer of potentials between the grounding area to outer points by buried conductors, such as communication or signal circuits, neutral wires, pipes, rails, or metallic fences, may produce serious safety problems. In recent years, the authors have developed a numerical technique based on the boundary element method for grounding analysis in uniform and layered soil models . First, the main highlights of this boundary element numerical approach are summarized, and next, a new methodology for the transferred potential analysis is presented. Finally, some examples by using the geometry of real grounding systems in different cases of transferred potentials are presented.

Why do computer methods for grounding analysis produce anomalous results

Grounding systems are designed to guarantee personal security, protection of equipment, and continuity of power supply. Hence, engineers must compute the equivalent resistance of the system and the potential distribution on the earth surface when a fault condition occurs. While very crude approximations were available until the 1970s, several computer methods have been more recently proposed on the basis of practice, semiempirical works and intuitive ideas such as superposition of punctual current sources and error averaging. Although these techniques are widely used, several problems have been reported. Namely, large computational requirements, unrealistic results when segmentation of conductors is increased, and uncertainty in the margin of error. A boundary element formulation for grounding analysis is presented in this paper. Existing computer methods such as APM are identified as particular cases within this theoretical framework. While linear and quadratic leakage current elements allow to increase accuracy, computing time is reduced by means of new analytical integration techniques. Former intuitive ideas can now be explained as suitable assumptions introduced in the BEM formulation to reduce computational cost. Thus, the anomalous asymptotic behavior of this kind of method is mathematically explained, and sources of error are rigorously identified.

Topology optimization of structures: A minimum weight approach with stress constraints

Sizing and shape structural optimization problems are normally stated in terms of a minimum weight approach with constraints that limit the maximum allowable stresses and displacements. However, topology structural optimization problems have been usually stated in terms of a maximum stiffness (minimum compliance) approach. In this kind of formulations, the aim is to distribute a given amount of material in a certain domain, so that the stiffness of the resulting structure is maximized (the compliance, or energy of deformation, is minimized) for a given load case. Thus, the material mass is restricted to a predefined percentage of the maximum possible mass, while no stress or displacement constraints are taken into account. In this paper, we analyze and compare both approaches, and we present a FEM minimum weight with stress constraints (MWSC) formulation for topology structural optimization problems. This approach does not require any stabilization technique to produce acceptable optimized results, while no truss-like final solutions are necessarily obtained. Several 2D examples are presented. The optimized solutions seem to be correct from the engineering point of view, and their appearence could be considered closer to the engineering intuition than the traditional truss-like results obtained by means of the widespread maximum stiffness (minimum compliance) approaches.

High order shape design sensitivity : a unified approach

Abstract Three basic analytical approaches have been proposed for the calculation of sensitivity derivatives in shape optimization problems. The first approach is based on differentiation of the discretised equations. The second approach is based on variation of the continuum equations and on the concept of material derivative. The third approach is based upon the existence of a transformation that links the material coordinate system with a fixed reference coordinate system. This is not restrictive, since such a transformation is inherent to FEM and BEM implementations. In this paper, we present a generalization of the latter approach on the basis of a generic unified procedure for integration in manifolds. Our aim is to obtain a single, unified, compact expression to compute arbitrarily high order directional derivatives, independent of the dimension of the material coordinates system and of the dimension of the elements. Special care has been taken on giving the final results in terms of easy-to-compute expressions, and special emphasis has been made in holding recurrence and simplicity of intermediate operations. The proposed scheme does not depend on any particular form of the state equations, and can be applied to both, direct and adjoint state formulations. Thus, its numerical implementation in standard engineering codes should be considered as a straightforward process. As an example, a second order sensitivity analysis is applied to the solution of a 3D shape design optimization problem.

Improvements in the treatment of stress constraints in structural topology optimization problems

Topology optimization of continuum structures is a relatively new branch of the structural optimization field. Since the basic principles were first proposed by Bendsoe and Kikuchi in 1988, most of the work has been dedicated to the so-called maximum stiffness (or minimum compliance) formulations. However, since a few years different approaches have been proposed in terms of minimum weight with stress (and/or displacement) constraints. These formulations give rise to more complex mathematical programming problems, since a large number of highly non-linear (local) constraints must be taken into account. In an attempt to reduce the computational requirements, in this paper, we propose different alternatives to consider stress constraints and some ideas about the numerical implementation of these algorithms. Finally, we present some application examples.

Resolution of computational aeroacoustics problems on unstructured grids with a higher-order finite volume scheme

Computational fluid dynamics (CFD) has become increasingly used in the industry for the simulation of flows. Nevertheless, the complex configurations of real engineering problems make the application of very accurate methods that only work on structured grids difficult. From this point of view, the development of higher-order methods for unstructured grids is desirable. The finite volume method can be used with unstructured grids, but unfortunately it is difficult to achieve an order of accuracy higher than two, and the common approach is a simple extension of the one-dimensional case. The increase of the order of accuracy in finite volume methods on general unstructured grids has been limited due to the difficulty in the evaluation of field derivatives. This problem is overcome with the application of the Moving Least Squares (MLS) technique on a finite volume framework. In this work we present the application of this method (FV-MLS) to the solution of aeroacoustic problems.

Numerical Simulation of Transferred Potentials in Earthing Grids Considering Layered Soil Models

Computing the potential distribution on the earth surface when a fault condition occurs is essential to assure the security of the grounding systems in electrical substations. This paper presents a numerical formulation for the analysis of transferred earth potentials in a grounding installation due to metallic structures or conductors in the surroundings of the grounding grid when a layered soil model is considered . This transference of potentials between the grounding area to distant points by buried conductors, such as communication or signal circuits, neutral wires, pipes, rails, or metallic fences, may produce serious safety problems. The authors have recently developed a numerical technique based on the Boundary Element Method for the analysis of transferred earth potentials in the case of uniform soil models . In this work, it is presented its generalization for a two layer soil model. Thus, the main highlights of the numerical approach are summarized and some examples by using the geometry of real grounding systems in different cases of transferred potentials considering different soil models are presented.

A boundary element formulation for the substation grounding design

Abstract A Boundary Element approach for the numerical computation of substation grounding systems is presented. In this general formulation, several widespread intuitive methods (such as Average Potential Method (APM)) can be identified as the result of specific choices for the test and trial functions and suitable assumptions introduced in the Boundary Element Method (BEM) formulation to reduce computational cost. While linear and parabolic leakage current elements allow to increase accuracy, computing time is drastically reduced by means of new completely analytical integration techniques and semi-iterative methods for solving linear equations systems. This BEM formulation has been implemented in a specific Computer Aided Design system for grounding analysis developed in the last years. The feasibility of this new approach is demonstrated with its application to a real problem.