One way to tackle uncertainties during the optimization is the robust design optimization (RDO) approach. Using RDO, mean and standard deviation of the objective are optimized. The stochastic quantities are computed using probabilistic methods. One approach is to approximate the objective function using a Taylor series and compute the stochastic quantities of the approximation analytically. The first-order second-moment method approximates the objective function by a first-order polynomial. Using the principal sensitivity approach, only two function evaluations are needed to evaluate the robust objective and its gardient. However, in many cases the first-order approximation of the objective is very inaccurate. As an alternative, the second-order fourth-moment method approximates the objective function by a second order Taylor polynomial, leading to more accurate results. Using the finite difference method to compute the required derivatives, the number of objective evaluations scales quadratically with the number of random variables. Using the adjoint method for only two additional systems of equations per random variable have to be solved additionally. However, this framework is highly intrusive and therefore needs full access to the finite element code. At the same time, it includes tensors that are large compared to the system matrix leading to large memory requirements. In this contribution an alternative framework for the second-order fourth-moment method is presented. It is non-intrusive and requires only four function evaluations per random variable. The memory requirements are similar to a deterministic optimization and it can be applied to any kind of optimization problem.