Large-scale computational mechanics remains limited by the cost of solving partial differential equations (PDEs) under realistic boundary conditions. Quantum computing offers a complementary paradigm for representing high-dimensional solution spaces, but most quantum PDE solvers remain far from engineering practice because boundary conditions are simplified, restricted to structured grids, or treated only in special cases. This paper critically reviews quantum approaches to numerical simulation, covering quantum annealing, gate-based Harrow-Hassidim-Lloyd (HHL) and post-HHL linear-system solvers, Hamiltonian-simulation and Schrödingerization methods, quantum physics-informed neural networks, and variational quantum algorithms (VQAs). For each family it assesses which equation and boundary-condition classes are suited, where the bottlenecks lie, and which problems remain open, and it then examines three issues that cut across all families: the end-to-end solution pipeline, the cost of expectation estimation, and the treatment of boundary conditions. The boundary-condition gap identified by the review is studied through a variational quantum finite element (VQA-FEM) case study that imposes Dirichlet, Neumann, and Robin conditions at arbitrary degrees of freedom by a set-to-zero stiffness correction measured with partial Pauli measurement (PPM), applied in noiseless statevector simulation to an Euler–Bernoulli beam, the National Agency for Finite Element Methods and Standards (NAFEMS) T4 heat-transfer benchmark, and the NAFEMS LE1 plane-stress benchmark. Two further studies locate the limits of this route. The variational solver, unlike a classical solve, leaves a small residual on eliminated degrees of freedom that the objective-value metric does not reveal, and the residual grows by two orders of magnitude if the textbook practice of setting their diagonal to 1 is followed. On the beam benchmark, finite-shot and depolarizing-noise studies show that stiffness-operator conditioning amplifies measurement uncertainty and noise, motivating preconditioning. The paper closes with a survey of quantum topology optimization and an assessment of the quantum-advantage horizon: logarithmic qubit scaling offers a memory advantage, but practical speedup depends on measurement cost, state preparation, trainability, conditioning, and error-corrected hardware.
In this paper, a coupled hyper-viscoelastic wavelet finite element model (WFEM) is proposed to investigate nonlinear shock wave formation and propagation in hyper-viscoelastic rods. The Mooney–Rivlin hyperelastic model and the generalized Maxwell viscoelastic model are combined to capture both finite deformation-induced nonlinearities and rate-dependent viscous dissipation. Numerical results demonstrate that the proposed WFEM achieves high precision and computational efficiency using significantly fewer elements than the conventional finite element method. A power-law scaling relationship is proposed to predict steady-state shock wave thickness without complicated and inefficient time-domain computations. Results show a higher scaling exponent for viscosity, which indicates that the shock front thickness is more sensitive to viscoelastic dissipation than to hyperelastic nonlinearity. A dimensionless shock steepening index is further proposed to construct a novel two-dimensional phase diagram, and it is found that wave evolution can be divided into four distinct regimes: viscoelastic-dominated, competition, weak shock wave, and hyperelastic-dominated. This phase diagram provides a framework for predicting shock wave behavior under varying material and loading conditions. Additionally, gradient variations in viscoelastic and hyperelastic parameters are analyzed. Uniformly and continuously increasing viscosity gradients significantly intensify the viscoelastic dispersion effect and accelerate energy dissipation, whereas decreasing gradients behave similarly to low-viscosity homogeneous rods. These findings demonstrate that the spatial distribution of material parameters can effectively control and attenuate shock wave energy. This work can provide an efficient numerical framework and practical design principles for advanced vibration isolation and impact protection devices.
Layered quasicrystal nanoplates combine the distinctive properties of quasicrystals with the design flexibility of laminated structures, making them attractive for micro- and nano-devices and functional thin-film systems. Their mechanical response is governed by the competition between nonlocal softening and strain gradient hardening, while imperfect interfaces further modify the transfer and coordination of phonon-phason fields between adjacent layers. Existing models usually treat these two scale effects separately or rely on ideal interface assumptions, and therefore cannot fully capture the coupled roles of scale effects, interfacial compliance, and stacking configuration. In this work, an exact analytical framework is developed for the bending, free vibration, and buckling of one-dimensional hexagonal quasicrystal layered nanoplates within nonlocal strain gradient theory. Interfacial compliance and scale effects are incorporated into a unified formulation. The conventional pseudo-Stroh formalism, originally developed for second-order systems, is extended to higher-order phonon-phason coupled quasicrystal media, and exact propagator matrix solutions are constructed to obtain closed-form analytical expressions for the static, dynamic, and stability responses of multilayered nanoplates. The results show that the nonlocal parameter l induces stiffness softening through spatial averaging, whereas the strain gradient parameter μ produces stiffness hardening by suppressing sharp strain variations. These two effects are competitive rather than simply additive, and l = μ can be regarded as an effective transition boundary between softening-dominated and hardening-dominated responses. The stacking configuration and interfacial compliance jointly regulate the structural stiffness, vibration characteristics, and stability of the nanoplates. This study provides a theoretical basis for the design, optimization, and application of multilayered quasicrystal nanostructures.
Fretting occurs when two contacting surfaces undergo micro-scale oscillatory displacements, resulting in severe stress gradients under partial slip conditions. These stress gradients play a critical role in the initiation of micro-cracks, which can ultimately lead to component failure. Consequently, accurate prediction of crack initiation is essential for preventing failure and improving component durability. In this context, the present study aims to predict the crack nucleation boundary of the Ti–6Al–4V titanium alloy under fretting conditions in a cylinder-flat contact configuration. The experimental loading conditions ( P_i , Q_ci^* ) associated with the initiation of a crack length of 10 m were taken from the work of Fouvry et al. A finite element (FE) model was used to predict the numerical nucleation boundary, while a coupled Smith–Watson–Topper (SWT)–Mindlin approach was applied to determine the analytical boundary. A recent theoretical solution for the stress components, derived by Vazquez et al., was used to validate the FE model. Both fixed and variable averaging volume methods were implemented within the numerical framework, and their effects on the predicted nucleation boundary were investigated. A significant discrepancy, with an error range of 58.8–61.8 P=400 N . However, deviation emerges and progressively increases as the normal load rises, reaching its highest value at P=1500 N , with an error of 9.7 P=850 N and by 3.1 P=1500 N .
This study investigates the dynamic response and energy absorption of aluminum foam sandwich tubes under lateral blast loading. Lateral blast experiments were conducted using a ballistic pendulum system to capture the fundamental deformation and measure midpoint deflections. A theoretical model based on a three-stage deformation mechanism was developed to predict the tubes responses. Key parameters examined include the inner and outer tubes wall thickness, foam core relative density and circumferential gradient configuration distribution and blast intensity. Results indicated that deformation localizes primarily in the upper half of the tube with symmetric patterns, where outer tubes experience larger deflections and circumferential buckling compared to inner tubes. The theoretical formulations considering circumferential membrane forces, bending, and core compression accurately capture the observed deflections. Increasing outer wall thickness significantly enhances blast resistance, specific energy absorption (SEA), and core energy absorption ratio. For the circumferential alternating gradient core, when the relative density of the core in the blast region was lower than that on the opposite side, the dimensionless midpoint deflection of the inner tube was the minimized, and the core achieve the highest energy absorption ratio. These findings provide valuable insights for the design and optimization of foam sandwich tubular structures under blast loading.
