In the proposed probabilistic topology optimization method, the minimum penalized weight design of the structure is subjected to probabilistic compliance and side constraints. The calculation of the compliance value is based on the assumption that the magnitude of the loads has uncertainties so that their joint normal distribution functions, mean values, and covariances are known and a minimum probability value is given. Applying an appropriate approximation, the original problem is replaced by a deterministic one. By the use of the optimality conditions, an updated iterative design formulation is presented. Several new topology design problems are calculated with randomly given loads.

This work presents an innovative framework for thermoelastic-plastic reliability-based topology optimization, tackling challenges related to material uncertainties, geometric imperfections, and variations in volume fractions. An enhanced Bi-directional Evolutionary Structural Optimization (BESO) method is developed. It integrates thermoelastic-plastic finite element analysis with stochastic reliability constraints to achieve robust and efficient structural designs under combined thermal and mechanical loading. The framework incorporates advanced modeling techniques, including temperature-dependent material properties, elasto-plastic behavior, and eigenmode-based imperfection modeling. A key innovation lies in formulating reliability constraints by treating volume fraction as a random variable to model material usage uncertainty. This ensures compliance with target safety indices. The proposed methodology is verified through detailed numerical examples, including steel beam and shell structures subjected to temperatures up to 800 °C. Results show that the probabilistic designs achieved up to 30% higher load-bearing capacity compared to deterministic ones and demonstrated improved stress distribution and thermal resilience. These enhancements confirm the method’s effectiveness in achieving optimal layouts that balance material efficiency, structural stability, and reliability.

A detailed numerical simulation of Colored Self-Compacting Concrete (CSCC) was conducted in this research. Emphasis was placed on an innovative calibration methodology tailored for ten unique CSCC mix designs. Through the incorporation of multi-objective optimization, MATLAB's Genetic Algorithm (GA) was seamlessly integrated with PFC3D, a prominent Discrete Element Modeling (DEM) software package. This integration facilitates the exchange of micro-parameter values, where MATLAB’s GA optimizes these parameters, which are then input into PFC3D to simulate the behavior of CSCC mix designs. The calibration process is fully automated through a MATLAB script, complemented by a fish script in PFC, allowing for an efficient and precise calibration mechanism that automatically terminates based on predefined criteria. Central to this approach is the Uniaxial Compressive Strength (UCS) test, which forms the foundation of the calibration process. A distinguishing aspect of this study was the incorporation of pigment effects, reflecting the cohesive behavior of cementitious components, into the micro-parameters influencing the cohesion coefficient within DEM. This innovative approach ensured significant alignment between simulations and observed macro properties, as evidenced by fitness values consistently exceeding 0.94. This investigation not only expanded the understanding of CSCC dynamics but also contributed significantly to the discourse on advanced concrete simulation methodologies, underscoring the importance of multi-objective optimization in such studies.

The aim of this paper is to propose a novel computational technique of applying reliability-based design to thermoelastic structural topology optimization. Therefore, the optimization of thermoelastic structures' topology based on reliability-based design is considered by utilizing geometrical nonlinearity analysis. For purposes of introducing reliability-based optimization, the volume fraction parameter is viewed as a random variable with a normal distribution having a mean value and standard deviation. The Monte Carlo simulation approach for probabilistic designs is used to calculate the reliability index, which is used as a constraint related to the volume fraction constraint of the deterministic problem. A new bi-directional evolutionary structural optimization scheme is developed, in which a geometrically nonlinear thermoelastic model is applied in the sensitivity analysis. The impact of changing the constraint of a defined volume of the required design in deterministic problems is examined. Additionally, the impact of altering the reliability index in probabilistic problems is investigated. The effectiveness of the suggested approach is shown using a benchmark problem. Additionally, this research takes into account probabilistic thermoelastic topology optimization for a 2D L-shaped beam problem.