Assessing the shakedown capacity of large-scale thin-walled structures under cyclic loading is essential for ensuring their long-term service safety. Existing direct shakedown methods are mostly based on a single type of finite element discretization, which leads to a persistent trade-off between computational efficiency and local accuracy for large-scale engineering structures: full solid models involve a prohibitive number of degrees of freedom, whereas pure shell models cannot adequately capture local three-dimensional stress gradients in regions such as joints and nozzle-shell intersections. To address this issue, this study proposes a lower-bound direct shakedown analysis framework for mixed-dimensional shell-solid structures according to which the linearized virtual nodes are constructed on the shell side, and an interface kinematic mapping matrix is established using radial basis function (RBF) interpolation. The corresponding static equilibrium conditions are derived from the principle of virtual work so as to accurately transfer generalized forces and moments across the interface. For the yield conditions, the von Mises criterion in the solid subdomain and the Ilyushin criterion in the shell subdomain are adopted. Both of them are formulated into standard second-order cone constraints, leading to a mixed-dimensional second-order cone programming (SOCP) model. The proposed method is systematically validated using a square plate with a hole, a three-dimensional pipe joint, and a pressure vessel with a nozzle. The results show that the shakedown limits predicted by the hybrid models agree well with those of full solid benchmarks, demonstrating the accuracy and engineering applicability of the proposed method. The framework provides an efficient and accurate analysis tool for shakedown assessment of large-scale thin-walled structures.
In the development of topology optimization methods for Additive Manufacturing (AM), accounting for the AM process-induced anisotropy of fabricated materials and the large deformations of slender members constitutes two significant engineering challenges. This paper proposes a novel structural topology optimization method that concurrently considers both effects. Firstly, the method introduces the print-off angle as an additional design variable, extending the orthotropic material model to characterize the anisotropic properties of AM-formed materials. Then, based on the finite deformation theory, the geometrically nonlinear mechanical behavior is simulated using the total Lagrangian formulation. The process-induced anisotropic constitutive relation is embedded into the geometrically nonlinear finite element equations, forming a coupled mechanical analysis framework that combines these two aspects. Furthermore, the Newton–Raphson method is applied to satisfy equilibrium conditions under large deformations. Additionally, numerical techniques, including density filtering, threshold projection, and adaptive control of the angular variable’s feasible domain, are integrated into the optimization process to stabilize the iterations. Finally, to enable efficient solutions via gradient-based optimization algorithms, sensitivity analyses with respect to both density and angle variables are derived in detail. Representative numerical examples demonstrate the effectiveness of the proposed approach and reveal characteristic features of the optimized designs. The results show that the process-induced anisotropy can influence geometrically nonlinear design, with its effect tending to intensify as anisotropy deepens. The extent of this influence, however, appears to be problem-dependent. Optimizing the printing direction may offer performance benefits, particularly when design conditions are complex.
Funicular shells carry loads predominantly through membrane action, achieving high structural efficiency with little material. Their computational design is difficult: the space of feasible forms is large, and classical optimisation explores it through local, problem-specific searches. Diffusion models offer an alternative by sampling from a learned distribution of plausible geometries, but coupling them with structural constraints is challenging, because physical guidance must act on noisy intermediate states whereas continuum-mechanical operators are defined on smooth geometries. Evaluating a non-linear shell operator on a denoised point estimate therefore yields a response that differs from the conditional mechanical response of the current diffusion state. We propose a modular two-branch Architect–Engineer pipeline. The Architect is a velocity-parameterised denoising diffusion model that learns an unconditional geometric prior over shell elevation maps. The Engineer is a time-conditioned Physics-Based Parallel UNet surrogate trained directly on noisy diffusion states, so that it predicts the mechanical response associated with the states visited during reverse sampling. The two models are trained independently and coupled only at inference, through a differentiable Membrane-Factor guidance pathway modulated by a bell-shaped schedule that concentrates the correction in the mid reverse process. On a dataset of funicular shells, the guided sampler raises the mean predicted Membrane Factor from 0.67 to 0.96, with 89
Three-dimensional (3D) viscoelastic wave modeling is essential in seismic applications, but its computational cost becomes prohibitive as domain size and simulation duration increase. To address this challenge, this study proposes a reduced-order modeling framework based on Proper Generalized Decomposition (PGD). A partially separated 2D + 1D representation decomposes the original 3D problem into coupled lower-dimensional subproblems. A high-order absorbing boundary condition based on the multi-transmitting formula (MTF) is embedded into the PGD framework, together with a singular value decomposition (SVD)-based recompression strategy that preserves the low-rank structure. Viscoelastic attenuation and point moment-tensor seismic sources are also incorporated. Numerical experiments are conducted on elastic full-space and half-space benchmarks, viscoelastic media spanning quality factors from purely elastic to strongly attenuating, multiple moment-tensor source mechanisms, and a horizontally layered half-space. The results demonstrate that the PGD-MTF effectively suppresses spurious boundary reflections and accurately reproduces the tested 3D wavefields in homogeneous and horizontally layered media. Performance benchmarks against the conventional spectral element method show that runtime and memory consumption are consistently reduced on the same benchmark mesh, with the gain depending on the scaling pattern. For the volume-scaling scenario at the largest tested mesh, the runtime speedup reaches 17.28, while the memory consumption is reduced by a factor of 33.98. Out-of-plane scaling further amplifies the advantage because it best matches the separated representation. The gain grows monotonically with problem size, indicating that PGD-MTF is an efficient and scalable tool for large-scale 3D seismic wave simulations in settings compatible with the adopted separated representation.
We propose a non-intrusive, Autoencoder-based framework for reduced-order modeling in continuum mechanics. Our method integrates three stages: (i) an unsupervised Autoencoder compresses high-dimensional finite element solutions into a compact latent space, (ii) a supervised regression network maps problem parameters to latent codes, and (iii) an end-to-end surrogate reconstructs full-field solutions directly from input parameters. To overcome limitations of existing approaches, we propose two key extensions: a force-augmented variant that jointly predicts displacement fields and reaction forces at Neumann boundaries, and a multi-field architecture that enables coupled field predictions, such as in thermo-mechanical systems. The framework is validated on nonlinear benchmark problems involving heterogeneous composites, anisotropic elasticity with geometric variation, and thermo-mechanical coupling. Across all cases, it achieves accurate reconstructions of high-fidelity solutions while remaining fully non-intrusive. These results highlight the potential of combining deep learning with dimensionality reduction to build efficient and extensible surrogate models.
