Ignasi Colominas

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 Numerical Formulation for Grounding Analysis in Stratified Soils

The design of safe grounding systems in electrical installations is essential to assure the security of the persons, the protection of the equipment, and the continuity of the power supply. To achieve these goals, it is necessary to compute the equivalent electrical resistance of the system and the potential distribution on the earth surface when a fault condition occurs. This paper presents a formulation for the analysis of grounding systems embedded in stratified soils, on the basis of the boundary element method (BEM). Suitable arrangements of the final discretized equations allow the use of highly efficient analytical integration techniques derived by the authors for grounding systems buried in uniform soils. Feasibility of this approach is demonstrated by applying the HEM formulation to the analysis of a real grounding system with a two-layer soil model.

Capillary networks in tumor angiogenesis: From discrete endothelial cells to phase-field averaged descriptions via isogeometric analysis†

Tumor angiogenesis, the growth of new capillaries from preexisting ones promoted by the starvation and hypoxia of cancerous cell, creates complex biological patterns. These patterns are captured by a hybrid model that involves high-order partial differential equations coupled with mobile, agent-based components. The continuous equations of the model rely on the phase-field method to describe the intricate interfaces between the vasculature and the host tissue. The discrete equations are posed on a cellular scale and treat tip endothelial cells as mobile agents. Here, we put the model into a coherent mathematical and algorithmic framework and introduce a numerical method based on isogeometric analysis that couples the discrete and continuous descriptions of the theory. Using our algorithms, we perform numerical simulations that show the development of the vasculature around a tumor. The new method permitted us to perform a parametric study of the model. Furthermore, we investigate different initial configurations to study the growth of the new capillaries. The simulations illustrate the accuracy and efficiency of our numerical method and provide insight into the dynamics of the governing equations as well as into the underlying physical phenomenon.

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.

Computer analysis of earthing systems in horizontally or vertically layered soils

Abstract The main objectives of grounding systems are to guarantee the personal safety, the equipment protection and the continuity of the power supply. To attain these goals, the potential distribution on the earth surface and the equivalent resistance of the earthing system have to be determined in a reliable and efficient way. In this paper, we present a numerical technique based on the Boundary Element Method (BEM) for grounding analysis of horizontally and vertically layered soils. The feasibility of the proposed method is demonstrated by means of its application to a real grounding grid considering different types of soil models.

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.

An efficient MP algorithm for structural shape optimization problems

Integral methods -such as the Finite Element Method (FEM) and the Boundary Element Method (BEL1)are frequently used in structural optimization problems to solve systems of partial differential equations. Therefore: one must take into account the large computational requirements of these sophisticated techniques at the time of choosing a suitable Mathematical Programming (MP) algorithm for this kind of problems. Among the currently available M P algorithms, Sequential Linear Programming (SLP) seems t o be one of the most adequate to structural optimization. Basically, SLP consist in constructing succesive linear approximations to the original non linear optimization problem within each step. However, the application of SLP may involve important malfunctions. Thus, the solution to the approximated linear problems can fail to exist, or may lead to a highly unfeasible point of the original non linear problem; also, large oscillations often occur near the optimum, precluding the algorithm to converge. In this paper, we present an improved SLP algorithm with line-search. specially designed for structural optimization problems. In each iteration) an approximated linear problem with aditional side constraints is solved by Linear Programming. The solution to this linear problem defines a search direction. Then, the objective function and the non linear constraints are quadratically approximated in the search direction, and a line-search is performed. The algorithm includes strategies t o avoid stalling in the boundary of the feasible region, and to obtain alternate search directions in the case of incompatible linearized constraints. Techniques developed by the authors for efficient high-order shape sensitivity analysis are referenced. Transactions on the Built Environment vol 52,

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.