. According to (Gomes , et al., 2013) the three main mathematical encounters with SIMP method are mesh dependency, checkboard patterns and local minima. To compensate for mesh-dependency and checkboard patterns instabilities, a well-known sensitivity filter brought forth by Bendose Sigmund is used (Gomes , et al., 2013). The local minima problem is dealt with by a continuation method with various increments in the function of the given filter (Gomes , et al., 2013). Pertaining to Bendose and Sigmund, the stiffness gained from SIMP model can be grasped as the stiffness of structures consisting of void and a quantity of solid material for the given density of the structure’s material (Zhu, et al., 2016). Because of the simplicity of SIMP*…show more content…*

The problems can be solved as void if ρ = 0 or associated material if ρ =1. Therefore, if ρ tends to be zero, the stiffness in that given element is zero, which means the element can be deleted because it is no longer important for the structure (Rao, et al., n.d.). However, if the density reaches one then that element is of dire importance to the structure and cannot be deleted (Rao, et al., n.d.). If this simple formulation is used, the total elastic energy measure (U) can be used as the objective function of the optimisation problem. This formulation is written below: Minimise: U n=1.....,N Subject to: ∑_(N-1)^N▒〖〖ρ〗_n V_n=V_0 〗 (2.1) 0 ≤ ρn ≤ 1 n = 1……,N This formulation in equation (1) and can also be extended to suite multiple load case problems by minimising the weight of the total elastic energies. The following equation expresses formulation for a multi-load case topology problem, using a weighted sum function for multi-load cases. Minimise: ∑_(j=1)^j▒〖w_j U_j 〗 Subject to: ∑_(N-1)^N▒〖ρ_n V_n=V_0 〗 (2.2) 0 ≤ ρn ≤ 1 n = 1,….,N Solving a Topology problem using this energy based minimum compliance topology optimisation

The problems can be solved as void if ρ = 0 or associated material if ρ =1. Therefore, if ρ tends to be zero, the stiffness in that given element is zero, which means the element can be deleted because it is no longer important for the structure (Rao, et al., n.d.). However, if the density reaches one then that element is of dire importance to the structure and cannot be deleted (Rao, et al., n.d.). If this simple formulation is used, the total elastic energy measure (U) can be used as the objective function of the optimisation problem. This formulation is written below: Minimise: U n=1.....,N Subject to: ∑_(N-1)^N▒〖〖ρ〗_n V_n=V_0 〗 (2.1) 0 ≤ ρn ≤ 1 n = 1……,N This formulation in equation (1) and can also be extended to suite multiple load case problems by minimising the weight of the total elastic energies. The following equation expresses formulation for a multi-load case topology problem, using a weighted sum function for multi-load cases. Minimise: ∑_(j=1)^j▒〖w_j U_j 〗 Subject to: ∑_(N-1)^N▒〖ρ_n V_n=V_0 〗 (2.2) 0 ≤ ρn ≤ 1 n = 1,….,N Solving a Topology problem using this energy based minimum compliance topology optimisation

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## Mesh Dependency And Simp Method

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This approach is called Solid Isotropic Material with Penalty (SIMP), in which material is assumed to be dependent upon density linearly. In accordance to the SIMP technique, the design variables are a sum of material densities denoted as (ρ) that relates the stiffness of an element straight to the density of that element (James , et al., 2014). According to (Gomes , et al., 2013) the three main mathematical encounters with SIMP method are mesh dependency, checkboard patterns and local minima. To

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