Research paper DOI 10.1007/s00158-003-0363-y Struct Multidisc Optim 27, 1–19 (2004) Structural shape and topology optimization in a level-set-based framework of region representation X. Wang, M.Y.Wang and D. Guo Abstract In this paper we present a new framework to approach the problem of structural shape and topology optimization. We use a level-set method as a region repre- sentation with a moving boundary model. As a boundary optimization problem, the structural boundary descrip- tion is implicitly embedded in a scalar function as its “iso- surfaces.” Such level-set models are flexible in handling complex topological changes and are concise in describ- ing the material regions of the structure. Furthermore, by using a simple Hamilton–Jacobi convection equation, the movement of the implicit moving boundaries of the struc- ture is driven by a transformation of the objective and the constraints into a speed function that defines the level- set propagation. The result is a 3D structural optimiza- tion technique that demonstrates outstanding flexibility in handling topological changes, the fidelity of boundary representation, and the degree of automation, compar- ing favorably with other methods in the literature based on explicit boundary variation or homogenization. We present two numerical techniques of conjugate mapping and variational regularization for further enhancement of the level-set computation, in addition to the use of effi- cient up-wind schemes. The method is tested with several examples of a linear elastic structure that are widely re- ported in the topology optimization literature. Key words structural optimization, topology optimiza- tion, shape optimization, boundary optimization, level- set methods, region representation Received: 2 August 2002 Revised manuscript received: 26 June 2003 Published online: 1 April 2004  Springer-Verlag 2004 X. Wang 1 , M.Y. Wang 2, ✉ and D. Guo 1 1 School of Mechanical Engineering, Dalian University of Tech- nology, Dalian 116024, China 2 Department of Automation & Computer-Aided Engineering, The Chinese University of Hong Kong, Shatin, NT, Hong Kong e-mail: yuwang@acae.cuhk.edu.hk 1 Introduction In this paper we address the problem of shape and top- ology optimization of a linearly elastic structure to meet a design objective and to satisfy certain constraints. The problem is formulated in a level-set framework in which the design domain is described by a structural bound- ary that is embedded in a scalar function of higher di- mensionality. As a level set or an “iso-contour” of the embedding function (also called the level-set function ), the boundary is implicitly described without the need of an explicit representation. The optimization process is captured by a Hamilton–Jacobi-type partial differential equation (PDE) that governs the dynamic movement of the embedding function and hence changes in the struc- tural boundary in accordance with the design objective and the constraints. While the shape and connectivity (i.e., topology) of the boundary may undergo drastic changes, the level-set function remains simple in its top- ology. Therefore, by a direct and efficient computation in the embedding space, the design boundaries can be tracked to a required level of accuracy, yielding an opti- mal structure in both shape and topology. The level-set models are referred to as a region representation and they can easily represent complex boundaries that can form holes, split into multiple pieces, or merge with others to form a single one. Based on the concept of propagation of the level-set interface, an optimization algorithm is de- rived from the shape sensitivity and the variations of the level-set embedded boundary. Boundary-based shape optimization has been a major method for structural design (Rozvany 1989; Sokolowski 1992). In essence, the design domain is directly repre- sented by its boundary, and a set of design variables di- rectly controls the exterior and interior boundary shapes, for example, through the control points of B-splines. Based on a boundary shape sensitivity analysis, neces- sary boundary variations for the optimality conditions would provide the foundation of an optimization tech- nique (Sokolowski 1992). It is a direct approach and it