Integration of Engineering Optimization in Architectural Design

Sakai Y., Ohsaki M., Adriaenssens S.

In this paper, we present a 3-dimensional elastic beam model for the form-finding and analysis of elastic gridshells subjected to bending deformation at the self-equilibrium state. Although the axial, bending, and torsional strains of the beam elements are small, the curved beams connected by hinge joints are subjected to large-deformation. The directions and rotation angles of the unit normal vectors at the nodes of the curved surfaces in addition to the translational displacements are chosen as variables. Based on the 3-dimensional elastic beam model, deformation of an element is derived from only the local geometrical relations between the orientations of elements and the unit normal vectors at nodes without resorting to a large rotation formulation in the 3-dimensional space. Deformation of a gridshell with hinge joints is also modeled using the unit normal vectors of the surface. An energy-based formulation is used for deriving the residual forces at the nodes, and the proposed model is implemented within dynamic relaxation method for form-finding and analysis of gridshells. The accuracy of the proposed method using dynamic relaxation method is confirmed in comparison to the results by finite element analysis. The results are also compared with those by optimization approach for minimizing the total potential energy derived using the proposed formulation.

He Y., Cai K., Zhao Z., Xie Y.M.

Topology optimization techniques have been widely used in structural design. Conventional optimization techniques usually are aimed at achieving the globally optimal solution which maximizes the structural performance. In practical applications, however, designers usually desire to have multiple design options, as the single optimal design often limits their artistic intuitions and sometimes violates the functional requirements of building structures. Here we propose three stochastic approaches to generating diverse and competitive designs. These approaches include (1) penalizing elemental sensitivities, (2) changing initial designs, and (3) integrating the genetic algorithm into the bi-directional evolutionary structural optimization (BESO) technique. Numerical results demonstrate that the proposed approaches are capable of producing a series of random designs, which possess not only high structural performance, but also distinctly different topologies. These approaches can be easily implemented in different topology optimization techniques. This work is of significant practical importance in architectural engineering where multiple design options of high structural performance are required.

Sundaram S., Skouras M., Kim D.S., van den Heuvel L., Matusik W.

Actuators of notable complexity are fabricated using multimaterial 3D printing coupled with automated design synthesis.

Yang K., Zhao Z., He Y., Zhou S., Zhou Q., Huang W., Xie Y.M.

Shape and topology optimization techniques are widely used to maximize the performance or minimize the weight of a structure through optimally distributing its material within a prescribed design domain. However, existing optimization techniques usually produce a single optimal solution for a given problem. In practice, it is highly desirable to obtain multiple design options which not only possess high structural performance but have distinctly different shapes and forms. Here we present five simple and effective strategies for achieving such diverse and competitive structural designs. These strategies have been successfully applied in the computational morphogenesis of various structures of practical relevance and importance. The results demonstrate that the developed methodology is capable of providing the designer with structurally efficient and topologically different solutions. The structural performance of alternative designs is only slightly lower than that of the optimal design. This work establishes a general approach to achieving diverse and competitive structural forms, which holds great potential for practical applications in architecture and engineering.

Chun J., Paulino G.H., Song J.

When conventional filtering schemes are used in reliability-based topology optimization (RBTO), identified solutions may violate probabilistic constraints and/or global equilibrium. In order to address this issue, this paper proposes to incorporate a discrete filtering technique termed the discrete filtering method (Ramos Jr. and Paulino 2016) into RBTO using the elastic formulation of the ground structure method. The discrete filtering method allows the optimizer to achieve more physically realizable truss designs in which thin bars are eliminated while ensuring global equilibrium. The method uses a potential-energy-based approach with Tikhonov regularization to solve the singular system of equations that may result from imposing the discrete filter. Combining this method with RBTO allows us to use the reliability-based truss sizing optimization for the purpose of topology optimization under uncertainties. Furthermore, a single-loop approach is adopted to enhance the computational efficiency of the proposed RBTO method. Numerical examples of two- and three-dimensional engineering designs demonstrate useful features of the proposed method and illustrate the influence of the discrete filter and parameter uncertainties on the optimization results. In order to check if the optimal topologies obtained by the proposed approach satisfy the constraints on the failure probabilities, structural reliability analysis is also performed using the first-order reliability method and Monte Carlo simulations.

Chun J., Song J., Paulino G.H.

For the purpose of reliability assessment of a structure subject to stochastic excitations, the probability of the occurrence of at least one failure event over a time interval, i.e. the first-passage probability, often needs to be evaluated. In this paper, a new method is proposed to incorporate constraints on the first-passage probability into reliability-based optimization of structural design or topology. For efficient evaluations of first-passage probability during the optimization, the failure event is described as a series system event consisting of instantaneous failure events defined at discrete time points. The probability of the series system event is then computed by use of a system reliability analysis method termed as the sequential compounding method. The adjoint sensitivity formulation is derived for calculating the parameter sensitivity of the first-passage probability to facilitate the use of efficient gradient-based optimization algorithms. The proposed method is successfully demonstrated by numerical examples of a space truss and building structures subjected to stochastic earthquake ground motions.

Baek C., Sageman-Furnas A.O., Jawed M.K., Reis P.M.

Significance Elastic gridshells arise from the buckling of an initially planar grid of rods. The interaction of elasticity and geometric constraints makes their actuated shapes difficult to predict using classical methods. However, recent progress in extreme mechanics reveals the benefits of structures that buckle by design, when exploiting underlying geometry. Here, we demonstrate the geometry-driven nature of elastic gridshells. We use a geometric model, originally for woven fabric, to rationalize their actuated shapes and describe their nonlocal response to loading. Validation is provided with precision experiments and rod-based simulations. The prominence of geometry in elastic gridshells that we identify should allow for our results to transfer across length scales from architectural structures to micro/nano–1-df mechanical actuators and self-assembly systems.

Martínez-Frutos J., Martínez-Castejón P.J., Herrero-Pérez D.

Osanov M., Guest J.K.

Advanced manufacturing processes provide a tremendous opportunity to fabricate materials with precisely defined architectures. To fully leverage these capabilities, however, materials architectures must be optimally designed according to the target application, base material used, and specifics of the fabrication process. Computational topology optimization offers a systematic, mathematically driven framework for navigating this new design challenge. The design problem is posed and solved formally as an optimization problem with unit cell and upscaling mechanics embedded within this formulation. This article briefly reviews the key requirements to apply topology optimization to materials architecture design and discusses several fundamental findings related to optimization of elastic, thermal, and fluidic properties in periodic materials. Emerging areas related to topology optimization for manufacturability and manufacturing variations, nonlinear mechanics, and multiscale design are also discussed.

Filipov E.T., Chun J., Paulino G.H., Song J.

We use versatile polygonal elements along with a multiresolution scheme for topology optimization to achieve computationally efficient and high resolution designs for structural dynamics problems. The multiresolution scheme uses a coarse finite element mesh to perform the analysis, a fine design variable mesh for the optimization and a fine density variable mesh to represent the material distribution. The finite element discretization employs a conforming finite element mesh. The design variable and density discretizations employ either matching or non-matching grids to provide a finer discretization for the density and design variables. Examples are shown for the optimization of structural eigenfrequencies and forced vibration problems.

Mueller C.T., Ochsendorf J.A.

This paper addresses the need to consider both quantitative performance goals and qualitative requirements in conceptual design. A new computational approach for design space exploration is proposed that extends existing interactive evolutionary algorithms for increased inclusion of designer preferences, overcoming the weaknesses of traditional optimization that have limited its use in practice. This approach allows designers to set the evolutionary parameters of mutation rate and generation size, in addition to parent selection, in order to steer design space exploration. This paper demonstrates the potential of this approach through a numerical parametric study, a software implementation, and series of case studies.

Bouhaya L., Baverel O., Caron J.

Gridshells are defined as structures that have the shape and rigidity of a double curvature shell but consist of a grid instead of a continuous surface. This study concerns those obtained by elastic deformation of an initially flat two-way grid. This paper presents a novel approach to generate gridshells on an imposed shape under imposed boundary conditions. A numerical tool based on a geometrical method, the compass method, is developed. It is coupled with genetic algorithms to optimize the orientation of gridshell bars in order to minimize the stresses and therefore to avoid bar breakage during the construction phase. Examples of application are shown.

Zhang L., Li Y., Cao Y., Feng X.

A highly efficient form-finding method of tensegrity is presented on the basis of the structural stiffness matrix, which is defined as the derivative of the out-of-balance force vector with respect to the nodal coordinate vector. The stiffness matrix and the total potential energy of the structure are utilized to direct the rapid convergence of the structural configuration to the self-equilibrated and stable state. In the programmed procedure, we employ the stochastic selecting algorithm to exclude rigid-body motions, the restricted step algorithm to guarantee the positive definiteness of the structural stiffness matrix, and the line search algorithm to minimize the total potential energy. This form-finding method allows us to easily determine the self-equilibrated and stable configuration of a tensegrity from an arbitrary initial state. A number of representative examples are given to demonstrate its accuracy and efficacy for both regular and irregular tensegrity structures of large scale.

Deaton J.D., Grandhi R.V.

Topology optimization is the process of determining the optimal layout of material and connectivity inside a design domain. This paper surveys topology optimization of continuum structures from the year 2000 to 2012. It focuses on new developments, improvements, and applications of finite element-based topology optimization, which include a maturation of classical methods, a broadening in the scope of the field, and the introduction of new methods for multiphysics problems. Four different types of topology optimization are reviewed: (1) density-based methods, which include the popular Solid Isotropic Material with Penalization (SIMP) technique, (2) hard-kill methods, including Evolutionary Structural Optimization (ESO), (3) boundary variation methods (level set and phase field), and (4) a new biologically inspired method based on cellular division rules. We hope that this survey will provide an update of the recent advances and novel applications of popular methods, provide exposure to lesser known, yet promising, techniques, and serve as a resource for those new to the field. The presentation of each method’s focuses on new developments and novel applications.

Richardson J.N., Adriaenssens S., Filomeno Coelho R., Bouillard P.

► A novel approach to the preliminary design of grid shell structures. ► 2-Phase approach: form finding and grid topology and nodal positions optimization. ► Demonstrate on case study minimizing the mass of 3 grid shells. This paper demonstrates a novel two-phase approach to the preliminary structural design of grid shell structures, with the objective of material minimization and improved structural performance. The two-phase approach consists of: (i) a form-finding technique that uses dynamic relaxation with kinetic damping to determine the global grid shell form, (ii) a genetic algorithm optimization procedure acting on the grid topology and nodal positions (together called the ‘grid configuration’ in this paper). The methodology is demonstrated on a case study minimizing the mass of three 24 × 24 m grid shells with different boundary conditions. Analysis of the three case studies clearly indicates the benefits of the coupled form-finding and grid configuration optimization approach: material mass reduction of up to 50% is achieved.

Fontalvo-García J.S., Moya-Olivares M.J., Gomez A., Villalba-Morales J.D.