# Upper Limb Orthoses: Integrating Topology Optimization and 3D Printing for Custom Fit and Function
## 2 Topology Optimization and Problem Formulation  
The formulation of a typical optimization problem comprises three essential elements [17,18]: (i) a vector of input data (known as design variables) that describe all potential system designs, (ii) one or more objective functions that establish the system's criteria for assessing the efficiency of each design, and (iii) a set of constraint functions that determine the feasibility of each design. After formulating the optimization problem, the objective of the optimization process is to identify the combination of the design variables that optimizes the objective functions to the greatest extent possible, while also satisfying the constraints. The general formulation of an optimization problem can be expressed mathematically with the following Equation (1):  
\[min_{x}f(x) \tag{1}\] \[s.t.\] \[g_{j}(x)\leq 0,\ j=1,2,...,m\] \[h_{k}(x)=0,\ k=1,2,...,p\] \[x=[x_{i}],\ i=1,2,...,n\ x_{i,lower}{\leq}x_{i}{\leq}x_{i,upper}\]  
where \(f(x)\) is the objective function to be optimized (minimized or maximized depending on the problem), \(g_{j}(x)\) and \(h_{k}(x)\) represent the inequality and equality constraint functions, respectively, and the vector \(x\) specifies the design variables. The design variables \(x\) are set up to configure the objective function \(f(x)\), ensuring they adhere to the constraints \(g(x)\) and \(h(x)\). The parameters \(x_{i,lower}\) and \(x_{i,upper}\) represent the minimum and maximum thresholds of the vector \(x\), respectively, while the subscripts \(j\) and \(k\) indicate the count of inequality and equality constraints, respectively. A standard TO problem can be formulated as follows:  
\[C(x)=F^{T}\cdot U(x)\] \[s.t.\] \[F=K\cdot U(x) \tag{2}\] \[V(x)/V_{0}=f\] \[0\leq x_{e}\leq 1\ \ \ \ldots e=1:n_{e}\]  
where the structural compliance value \(C(x)\) is the objective function to be minimized, vector \(F\) encompasses the external forces applied, and \(U(x)\) refers to the displacement vectors calculated by solving the structural equilibrium equation \(F=K\cdot U(x)\). The variables \(V(x)\) and \(V_{0}\) represent the volume of the structure and the total domain volume, respectively, while \(n_{e}\) specifies the number of finite elements (FEs) used to discretize the structure. The target volume percentage is symbolized by \(f\), and \(x_{e}\) indicates the material density for element \(e\). In general, TO can be considered as the procedure of optimally distributing a specified material volume inside a specific design domain in order to transfer the loading conditions to the structural supports. The change in the material distribution is achieved by removing or adding material to areas of the design domain [19]. In contrast to sizing and shape optimization formulations which utilize dimensions or coordinates as design variables, TO introduces a new artificial quantity that defines the existence or absence of material within each FE, so-called relative material density. Thus, in the TO problem, the optimization algorithm mainly uses the material densities to indicate the existence of parts of the domain in order to achieve the final solution. The solution of the TO problem follows the Solid Isotropic Microstructure with Penalization for intermediate densities (SIMP) method [20; 21; 22]. SIMP is a gradient-based method based on a continuously converging algorithm, which is controlled by a penalization parameter, p. In the original design domain, each FE of the discretized initial section corresponds to a design variable, which is generally thought of as the relative element density, \(\rho_{e}\).