Allying topology and shape optimization through machine learning algorithms

https://doi.org/10.1016/j.finel.2021.103719Get rights and content

Highlights

  • We proposed a methodology that allies topology and parametrized shape optimization algorithms, obtaining an hybrid optimization algorithm.

  • We studied the strengths and weaknesses of topology and shape optimization algorithms, which justified its combination to obtain a synergetic effect.

  • The use of manifold learning techniques, such as the Locally Linear Embedding (LLE), to extract the principal geometrical modes from a set of material distributions that topology optimization algorithm provides.

  • In order to properly use the LLE technique, the quasi-boolean material distribution provided by the topology optimization algorithm must be transformed into a richer level-set with the distance information.

  • The geometrical modes, extracted with the LLE algorithm, may be used to generate brand new geometries, as they define a parametric geometrical model.

  • The parametric space of geometrical modes may be considered by a shape optimization algorithm as its design space.

Abstract

Structural optimization is part of the mechanical engineering field and, in most cases, tries to minimize the overall weight of a given design domain, subjected to functionality constraints given in terms of stresses or displacements. The most relevant techniques are topology and shape optimization. Topology optimization provides the optimal material distribution layout into a given, static, design domain. On the other hand, shape optimization provides the optimal combination of the parameters that define the required parametrization of the domain's boundary. Both techniques have strengths and weaknesses, thus a hybrid optimization approach that combines the former techniques will define a more general structural optimization framework that will take advantage of their synergistic combination. The difficulty arises when communicating both techniques for which, in this paper, we propose a machine learning-based methodology.

Introduction

Optimization is the mathematical discipline that tries to find the best element of a given set. The search is driven by the performance of each element, measured through a predefined loss function. Optimization techniques are extensively used in fields such as science, engineering or economics. We will focus on the engineering field, specially on structural optimization. Structural optimization is a crucial tool in the design process of mechanical components, since it is able to generate the optimal design domain according to a set of applied loads. The optimal design must minimize or maximize an objective function while satisfying a set of constraints. The most common pairs of objective function and constraints found in the structural optimization field are: the minimization of the mass/volume while satisfying a yielding stress constraint and the maximization of the stiffness while satisfying a volume fraction constraint. There exist different approaches to solve the structural optimization problems. Among them, we will focus on the most common ones, namely, topology and shape optimization techniques.

Topology optimization algorithms allow to modify the topology of the material in the design space at the expense of a large amount of design variables, such as the relative density of each element with the SIMP method [[1], [2], [3]], the distance of each node to the implicit boundary with the Level-Set approaches [[4], [5], [6]] or the Phase Field for topology optimization [[7], [8], [9]]. The current work is based on the SIMP method that provides an optimal material distribution layout over the design domain defined by a blurred boundary which is not directly suitable for manufacturing. A review of the SIMP method can be found in Appendix A.1. On the other hand, shape optimization techniques, use a CAD representation of the boundary of the geometry to compute the objective function and constraints. This CAD representation may be defined using many types of geometrical entities (splines, NURBS, etc.). In our case, the boundary will be represented using the STL format, i.e., a triangular tessellation of the geometric boundary. Thus, the optimal geometry provided is directly suitable for manufacturing. In this work, we consider the parameterized shape optimization algorithm which needs a user-defined parameterized boundary of fixed topology that does not allow to explore new topologies. The main benefit of using shape optimization techniques is the accuracy and smoothness of the boundary definition. This benefit is even greater if we take into account that the number of design variables necessary to parameterize the boundary of the geometry is usually low, this allowing the exploration of the design space with a huge variety of optimization algorithms. A description of the shape optimization problem considering geometrical parameters can be found in Appendix A.2.

Given the characteristics of these two types of optimization techniques, it would be desirable to develop a hybrid approach that harness the strengths and discards the weaknesses of the topology and shape optimization techniques when used separately.

This hybrid algorithm could be defined by the following steps:

  • 1.

    Topology optimization. This step should provide a preform with topological characteristics defined in terms of an optimal material distribution layout, consider the design domain defined by the analyst.

  • 2.

    Interface. This step should communicate both topology and shape optimization algorithms. This interface should generate the parametric geometrical model (defined by design variables) required by the shape optimization algorithm from the results of the topology optimization process.

  • 3.

    Shape optimization. The shape optimization algorithm should then use this model and will find the optimal combination of its parameters that minimize a given objective function while satisfying the prescribed constraints. The final results of this step should be a CAD-like representation of the optimal geometry directly suitable for manufacturing.

The main issue that we face when implementing such a hybrid algorithm is the development of step 2, the interface step, that allies both topology and shape optimization algorithms. Below are, to the authors' knowledge, the main contribution to this topic that can be found in the bibliography. Reference [10] manually parametrized the result of the topology optimization solution, and used it in the parametrized shape optimization algorithm. In the approach described in Refs. [11,12], the authors parametrize the optimal material distribution of a 2D design domain by means of curve fitting algorithms. The parameters that define those curves are then modified by the shape optimization algorithm to find the optimal geometry. Also, we would like to highlight the work in Ref. [13] where the use of Artificial Neural Networks allow to find the set of simple entities that reproduce the material distribution provided by the topology optimization algorithm. Additionally, alternative approaches are found in the literature. For instance, in Ref. [14] the authors use an edge detection technique to identify the structural elements provided by the topology optimization algorithm. In Ref. [15] the authors use a edge detection technique, the Canny algorithm, and manually create a B-spline representation of the model. In Ref. [16] the authors manually create the mesh for the shape optimization algorithm by means of the material distribution layout indicated by the topology optimization algorithm. Also, in Ref. [17] the authors simultaneously evaluate both optimization algorithms; in this case the shape is modified considering the variation of the nodal coordinates of the mesh by means of weights, acting as design variables, and predefined perturbation vectors. In Refs. [18,19], the authors create a two-stage algorithm where the overall geometric definition is achieve in the topology optimization step. Then the result is represented with Deformable Simplicial Complex entities whose vertices’ positions can be modified by the shape optimization algorithm. Also [20], presented a new level-set algorithm that allows to reduce the dimension of the functional by means of the Radial Basis Functions. Finally, we highlight the interesting work developed in Ref. [21] where the authors propose to first use a shape optimization algorithm to define the design domain and then to use a topology optimization algorithm to find the optimal material distribution.

In our work, we propose the use of a Machine Learning (ML) technique to infer the geometrical characterization defined by a set of parameters. Specifically, we use a Dimensionality Reduction (DR) algorithm, a subfield of the ML techniques. These algorithms will automatically create a parametric model, defined as a combination of geometrical modes that explicitly characterizes the implicit boundary given by the material distribution provided by the topology optimization algorithm. The extracted geometrical features may take the form of simple geometrical entities, such as radius or thickness but, in general, the extracted geometrical features will be more complex. In any case, the ML tool will be able to identify the geometrical modes, providing a parametric geometrical representation. We will then be able to use this parametric characterization to generate new geometries by modifying the value of the parameters, either manually or guided by an external algorithm. In our case, we will introduce these parameters as the design variables used by a shape optimization algorithm.

The paper is organized as follows. The implementation of the hybrid optimization algorithm relies on a set of technologies or methods, that are presented in Section 2. Following, in Section 3, we will describe the benchmark analytic problem used to check the functionalities developed. Then, in Section 4 we will describe the strategy considered to achieve an hybrid optimization framework and how the previously described technologies interact with each other. Later, in Section 5 we will show the behaviour of the proposed methodology by means of numerical analyses considering the benchmark problem together with the numerical analyses on the well-known MBB beam problem and a hook problem. Finally, in Section 6, we will conclude the paper with some final remarks. An Appendix is also included, for the sake of completeness, to properly describe some methods and technologies discussed in this work.

Section snippets

Methodologies

As proposed in Section 1, our goal is to ally topology and shape optimization techniques in order to develop a hybrid optimization framework. To accomplish this objective, we will make use of different methodologies and technologies.

On one hand, we will harness the capabilities of Machine Learning (ML) techniques to infer information from datasets.

The topology optimization technique produces intermediate solutions during the iterative process. After the initial steps of the process,

Reference benchmark problem

The reference problem used to describe the proposed methodologies is defined in Fig. 1 where we use a coherent system of units. This problem corresponds to a constant hollowed cross sectional area beam, with 2 perpendicular planes of symmetry (x = 0 and z = 0), under plain strain conditions, subjected to a pressure P on the internal cylindrical surface.

The objective of the optimization problem is to minimize the amount of material while the maximum von Mises stress value (max(σvm)) is equal to

Hybrid optimization

In this section we will describe the details about the strategy that we propose to ally topology and shape optimization techniques and to create a general structural optimization framework. We will describe the procedure developed to automatically extract the geometrical parameters and how we use them. In Fig. 2, we show the main steps in the proposed strategy.

To summarize, as represented in Fig. 2, we postprocessed the material distribution layout provided by the TO algorithm to create a

Numerical examples

After having shown the results provided by the proposed hybrid optimization algorithm for the academic reference problem, in this Section, we will use two additional examples. Specifically, we will use the well-known MBB problem and a hook problem.

Conclusion

In order to conclude the current work, we would like to synthesize some final remarks:

  • Shape optimization algorithms based on parametrized geometrical representations provide manufacturable solutions but are unable to explore topologies other that the topology described by the prescribed parametrized geometrical model.

  • The topology optimization algorithms based on the extensively used SIMP method for structural optimization provide a material distribution that characterizes the topology of the

Availability of data

Data available on request from the authors. The data that supports the findings of this study are available from the corresponding authors upon reasonable request.

Author statement

D. Muñoz: Methodology, Software, Validation, Investigation, Data Curation, Writing - Original Draft. E. Nadal: Conceptualization, Methodology, Investigation, Writing - Review & Editing. J. Albelda: Conceptualization, Methodology. F. Chinesta: Conceptualization, Writing - Review & Editing. J.J. Ródenas: Conceptualization, Methodology, Writing - Review & Editing, Supervision, Funding acquisition.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors gratefully acknowledge the financial support of Ministerio de Educación y Formación Profesional (FPU16/07121), Generalitat Valenciana (Prometeo/2021/046), Ministerio de Economía, Industria y Competitividad (DPI2017-89816-R) and FEDER.

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