Shape optimization of elastic structures is a popular area used in various industries and fields. Several methods have been developed to address this problem. In this paper, the level set method is studied. We observe shape optimization in 2D and model the problem in linearized elasticity. We set up a linearized elasticity system for an elastic body on which external forces act. The solution of such a system gives us the displacement of the elastic body. We maximize body stiffness which is represented by minimizing the target functions over the allowable domains. We solve the minimization problem with a gradient descent method that requires the introduction of a definition of shape derivative. We list various results that allow us to obtain expressions for shape derivatives of some objective functions. From the shape derivative we easily obtain the velocity of the moving part of shape boundary, we choose the direction of the descent in the gradient descent method and thus we have the general procedure of the algorithm. The shape boundary can be represented in a variety of ways from which we select level sets by which we have simpler expressions for normal and mean curvature. We extend the system of linearized elasticity to the entire working domain by the simple \(ersatz\) material approach. The level set function is obtained by solving the transport Hamilton–Jacobi equation. We propose an iterative algorithm in which we describe in more detail the procedures of initialization and iteration through the algorithm. We introduce and describe the procedure of re-initialization of the level set function and the procedure of volume restriction. In the second chapter we present the results obtained in the software tool FreeFem++. We list used parameters and some useful derivations required for implementation. We observe different examples and objective functions. The results are presented on the examples of 2D cantilever and 2D bridge, where we analyze them in more detail.