complex variables method

On the mechanics of solids, in many cases, the body is affected by mechanical force and heat load simultaneously. One of the most widely used and important parameters in the optimal design of engineering problems is the utilization of the sensitivity numbers with respect to the values of its design variables. The present paper analyzes sensitivity in thermomechanical problems using the complex variables method. The most important drawback of past methods for the analysis of sensitivity values such as finite difference and semi-analytical method is the dependence of the calculated values of sensitivity numbers on the step size, which leads to unreliable answers. Complex variables method (CVM) is a new robust approach that uses a Taylor series extension in a complex space and which is not sensitive to step size, disturbance and given that during the computational process, subtraction of two numbers is almost equally avoided, so does not influence from error caused by removing meaningful figures that commonly occur in finite difference method due to subtracting of two near the number. As a result, it is possible to achieve accurate answers by choosing a small step size. Also, it has some potential advantages over other methods. At first order sensitivities using the complex variables method, the implementation is straightforward, only requiring a perturbation of the finite element mesh along the imaginary axis. Implementation of sensitivity using CVM requires complex variable finite element code such that complex nodal coordinates can be used to implement a perturbation in the shape of interest in the complex domain. All resulting finite element outputs such as temperature, displacements, strains, stresses, etc. become complex and accurate derivatives of all finite element outputs to the shape parameter of interest are available. So, unlike most methods, using the proposed method does not require calculations to select the appropriate change in step size. However, the value of the change in step size is not known in advance and changes from one problem to another Therefore, it is necessary to repeat the simulation to ensure the results. The study found that the source of the rounding error was due to the rotation of the element, this also makes it difficult to use two methods, semi-analytical and finite-difference. In this case, the use of CVM is not limited and can be achieved by effective answers without worrying about the small step size. In the present study, to examine how efficient and valid the proposed method is, how it works to solve several thermomechanical problems is described. The methodologies are demonstrated using two-dimensional finite element models. Obtained sensitivity derivatives are compared to the semi-analytical solution and also finite-difference solution and it is shown that the proposed method is effective and can predict stable and accurate sensitivity results. The complex variable method presented in this study can be used to solve a wide range of engineering problems.