Optimization of injection molding process parameters is essential for improving product quality and production efficiency. Traditional methods, which rely heavily on operator experience, often result in inconsistencies, high time consumption, high defect rates, and suboptimal energy consumption. In this study, an improved particle swarm optimization (IPSO) algorithm was proposed, integrating dynamic inertia weight adjustment, adaptive acceleration coefficients, and position constraints to address the issue of premature convergence and enhance global search capabilities. A dual-model architecture was implemented: a constraint validation mechanism based on support vector machine (SVM) was enforced per iteration cycle to ensure stepwise quality compliance, while a fitness function derived by extreme gradient boosting (XGBoost) was formulated to minimize cycle time as the optimization objective. The results demonstrated that the average injection cycle time was reduced by 9.41% while ensuring that the product was qualified. The SVM and XGBoost models achieved high performance metrics (accuracy: 0.92; R²: 0.93; RMSE: 1.05), confirming their robustness in quality classification and cycle time prediction. This method provides a systematic and data-driven solution for multi-objective optimization in injection molding, significantly improving production efficiency and energy utilization.

The two-step backward differential formula (BDF2) with unequal time-steps is applied to construct an energy stable convex-splitting scheme for the Cahn-Hilliard model. We focus on the numerical influences of time-step variations by using the recent theoretical framework with the discrete orthogonal convolution kernels. Some novel discrete convolution embedding inequalities with respect to the orthogonal convolution kernels are developed such that a concise $L^2$ norm error estimate is established at the first time under an updated step-ratio restriction $0 <r_k:=\tau_k/\tau_{k-1}\leq r_{\mathrm{user}}$, where $r_{\mathrm{user}}$ can be chosen by the user such that $r_{\mathrm{user}}<4.864$. The stabilized convex-splitting BDF2 scheme is shown to be mesh-robustly convergent in the sense that the convergence constant (prefactor) in the error estimate is independent of the adjoint time-step ratios. The suggested method is proved to preserve a modified energy dissipation law at the discrete levels if $0<r_k\le r_{\mathrm{user}}$, such that it is mesh-robustly stable in an energy norm. On the basis of ample tests on random time meshes, a useful adaptive time-stepping strategy is applied to efficiently capture the multi-scale behaviors and to accelerate the long-time simulation approaching the steady state.

We propose and analyze a linearly stabilized semi-implicit diffusive Crank--Nicolson scheme for the Cahn--Hilliard gradient flow. In this scheme, the nonlinear bulk force is treated explicitly with two second-order stabilization terms. This treatment leads to linear elliptic system with constant coefficients and provable discrete energy dissipation. Rigorous error analysis is carried out for the fully discrete scheme. When the time step-size and the space step-size are small enough, second order accuracy in time is obtained with a prefactor controlled by some lower degree polynomial of $1/\varepsilon$. {Here $\varepsilon$ is the thickness of the interface}. Numerical results together with an adaptive time stepping are presented to verify the accuracy and efficiency of the proposed scheme.