This paper discusses optimal shape design as a material distribution problem. The goal is to determine the optimal spatial material distribution for given loads and boundary conditions. Each point in space is either material or void, making the problem a discrete variable optimization. To remove this discrete nature, a continuous density function is introduced, where high density defines the shape of the mechanical element. Intermediate densities can be handled using artificial material laws or homogenization through periodic voids. The paper introduces a general formulation for optimal shape design of linearly elastic structures. The problem is defined by an elasticity tensor that varies over the domain, with the material density function X(x) indicating whether a point is material or void. This formulation leads to a distributed parameter optimization problem with a discrete valued parameter function. Direct optimization using finite elements would require discrete algorithms, which are unstable without composite materials. The paper compares different methods for material distribution, including composites with square and rectangular voids, and layered media with weak materials as voids. It also presents results from an artificial power-law for rigidity dependence on density, showing how variable thickness sheets can be designed. The methods allow determination of the topology of a mechanical element and provide information on the optimal shape's boundary form. The paper also discusses the limitations of boundary variation techniques, which can only predict the optimal shape of a given initial topology. A new method that can determine both optimal topology and shape is proposed as an extension. The formulation of shape design problems as pointwise material/no material problems was introduced by Kohn and Strang, and the practical possibilities were studied by Bendsøe and Kikuchi. The use of composite materials allows the problem to be transformed into a material distribution problem, enabling the use of homogenization theory to compute effective material properties. The paper concludes with a general problem formulation for optimal shape design, emphasizing the importance of material distribution in achieving optimal structural layouts.This paper discusses optimal shape design as a material distribution problem. The goal is to determine the optimal spatial material distribution for given loads and boundary conditions. Each point in space is either material or void, making the problem a discrete variable optimization. To remove this discrete nature, a continuous density function is introduced, where high density defines the shape of the mechanical element. Intermediate densities can be handled using artificial material laws or homogenization through periodic voids. The paper introduces a general formulation for optimal shape design of linearly elastic structures. The problem is defined by an elasticity tensor that varies over the domain, with the material density function X(x) indicating whether a point is material or void. This formulation leads to a distributed parameter optimization problem with a discrete valued parameter function. Direct optimization using finite elements would require discrete algorithms, which are unstable without composite materials. The paper compares different methods for material distribution, including composites with square and rectangular voids, and layered media with weak materials as voids. It also presents results from an artificial power-law for rigidity dependence on density, showing how variable thickness sheets can be designed. The methods allow determination of the topology of a mechanical element and provide information on the optimal shape's boundary form. The paper also discusses the limitations of boundary variation techniques, which can only predict the optimal shape of a given initial topology. A new method that can determine both optimal topology and shape is proposed as an extension. The formulation of shape design problems as pointwise material/no material problems was introduced by Kohn and Strang, and the practical possibilities were studied by Bendsøe and Kikuchi. The use of composite materials allows the problem to be transformed into a material distribution problem, enabling the use of homogenization theory to compute effective material properties. The paper concludes with a general problem formulation for optimal shape design, emphasizing the importance of material distribution in achieving optimal structural layouts.