A new formulation is presented for the certified reduced-basis (RB) analysis of parametrized laminated composite plates by a finite element/polyharmonic spline-radial basis function-finite difference (FEM/PHS-RBF-FD) hybrid method. The full-order (“truth”) model is first-order shear deformation theory. The mid-surface is meshed with Q9 quadrilateral elements, but the Lagrange interpolation is replaced inside each element by PHS-RBF cardinal functions built from r^5 plus complete quadratic polynomials on the element node set, which keeps standard finite-element assembly and boundary handling while removing the RBF shape parameter. The plies are grouped by orientation and each group g carries a scalar stiffness multiplier μ _g , so the truth stiffness is exactly affine in the parameter, K(μ)=∑ _g=1^Gμ _g K_g , with symmetric positive semidefinite, precomputable components. This affine structure is exploited by a weak-greedy reduced-basis method with full offline–online decomposition and a rigorous a posteriori error bound: because every K_g≽ 0 and the energy inner product is chosen at the reference parameter, the min- θ coercivity lower bound is available in closed form and the dual norm of the residual is evaluated online from precomputed Gram matrices. The method is assessed on simply supported and clamped plates with cross-ply and angle-ply stacking sequences. A locking study shows that full integration of the shear term is adequate for a/h≤ 100 but locks for very thin plates ( a/h≳ 10^3 ), where a 2× 2 rule for the shear term is needed. The online stage is two to four orders of magnitude faster than a truth solve.
Using complex variable methods, we derive a closed-form solution to the two-dimensional and time-dependent Eshelby’s problem associated with a circular Eshelby inclusion undergoing uniform in-plane eigenstrains with rate-dependent interface slip and diffusion concentrically embedded in an isotropic elastic finite circular domain with a rigidly clamped or traction-free boundary. Explicit expressions for the time-dependent interior and exterior Eshelby tensors in the circular inclusion and in its supplement to the finite circular domain are presented. The stresses and strains within the circular inclusion are spatially non-uniform and decay with two relaxation times.
When applying the deep energy method (DEM) to highly heterogeneous multi-material systems, the inherent ‘spectral bias’ of neural networks leads to a preference for low-frequency signals, causing a spectral aliasing problem that results in excessive smoothing of the displacement field and making it difficult to resolve abrupt changes in interfacial stress. To address this, this paper proposes a physics-driven attention-enhanced DEM algorithm. This method prevents spectral aliasing through decoupled mapping of geometric and physical features and introduces stress-driven attention gating to enhance feature representation in regions with constitutive discontinuities and high-stress concentrations. A validation model of an aircraft tire, featuring 17 material layers with extreme stiffness contrasts and undergoing large hyperelastic deformation, is established using experimentally fitted data. Experiments demonstrate that the algorithm accurately reconstructs the mechanical transfer mechanisms at complex interfaces, achieving high-fidelity meshless solutions while matching the accuracy of commercial finite element software. Owing to its mesh-free nature, the DEM proposed in this paper possesses a natural advantage over mesh-based methods in circumventing convergence issues caused by severe mesh distortion. It provides a reliable new paradigm for high-fidelity mechanical simulation of complex composite structures and advances the implementation of deep learning-based partial differential equation solvers in industrial-scale applications.
We analyze how nonlocal stresslet interactions generate modal instability and finite-time nonmodal amplification in a particle-resolved zero-Reynolds-number active suspension. Three configuration-dependent operators are compared side by side: the orientation-projected pairwise interaction matrix W ( q ) , its activity-weighted counterpart L _p( q ,A)=A W ( q ) , and the full tangent Jacobian J ( q )=D_ q F ( q ) of the deterministic drift, constructed by centered finite differences at snapshots on an H-weighted state space. Within the sampled parameter range, once the direct activity scaling is divided out, the projected operator provides no incremental information beyond the imposed activity ramp. By contrast, the full tangent operator contains spatial derivatives, orientational derivatives, and translation–orientation coupling that generate substantial finite-time amplification in addition to modal growth: the pseudospectral abscissa a_ε =10^-1( J _H) exceeds the spectral abscissa a( J ) by 0.1–0.2, the eigenvector condition number κ (V) is of order 10^2 – 10^3 , and the normalized nonmodal gain R_nm(T=3) reaches ≈ 4.5 at the largest sampled activity. A logarithmic norm bound obtained through Grönwall’s inequality, ‖ (t_0+T,t_0)‖ _2^2≤exp[2∫ _t_0^t_0+Tω ( J _H(s)) ds] , transfers the frozen-time numerical abscissa into a rigorous upper bound on the trajectory propagator. The sampled regime is modally unstable within the parameter range studied; a stable-nonnormal regime a<0<ω is not exhibited by the simulations, and a weakly nonlinear normal form analysis is retained only as a formal outlook. The main limitation of the present study is that all numerical results concern a denominator-regularized minimum-image model rather than a fully smooth periodic Ewald Stokes formulation.
