Computational Mechanics最新文献

Failure initiation and subsequent crack trajectory in heterogeneous materials, such as functionally graded materials and bones, are yet insufficiently addressed. The AT1 phase field model (PFM) is investigated herein in a 1D setting, imposing challenges and opportunities when discretized by h- and p-finite element (FE) methods. We derive explicit PFM solutions to a heterogeneous bar in tension considering heterogeneous E(x) and (G_{Ic}(x)), necessary for verification of the FE approximations. (G_{Ic}(x)) corrections accounting for the element size at the damage zone in h-FEMs are suggested to account for the peak stress underestimation. p-FEMs do not require any such corrections. We also derive and validate penalty coefficient for heterogeneous domains to enforce damage positivity and irreversibility via penalization. Numerical examples are provided, demonstrating that p-FEMs exhibit faster convergence rates comparing to classical h-FEMs. The new insights are encouraging towards p-FEM discretization in a 3D setting to allow an accurate prediction of failure initiation in human bones.

A computational homogenization framework is presented to study the dynamics of locally resonant acoustic metamaterial structures. Modelling the resonant units at the microscale as representative volume elements and building on well-established scale transition relations, the framework brings as a main novelty a reduced-order macroscopic homogenized continuum whose governing equations involve no additional variables to describe the microscale dynamics unlike micromorphic homogenized continua obtained by alternative computational homogenization approaches. This model-order reduction is obtained by formulating the governing equations of the micro- and macroscale problems in the frequency domain, introducing a finite-element discretization of the two problems and performing an exact dynamic condensation of all the degrees of freedom at the microscale. An appropriate inverse Fourier transform approach is implemented on the frequency-domain equations to capture transient dynamics as well; notably, the implementation involves the Exponential Window Method, here applied for the first time to calculate the time-domain response of undamped locally resonant acoustic metamaterial structures. The framework may handle arbitrary geometries of micro- and macro-structures, any transient excitations and any boundary conditions on the macroscopic domain.

In this paper, we develop a Virtual Clustering Analysis method for a phase field model of brittle fracture. In addition to the strain/stress field, we treat the phase field variable via clusters as well, based on Green’s function of the governing Helmholtz equation. Around the crack path, we assign one cluster per cell. We detect the crack tip and recluster accordingly as the crack propagates. Three examples are presented to demonstrate the numerical efficiency of the proposed method, including either straight or curved crack, under tension or shear.

The frequency solutions of finite elements may significantly deteriorate as the mesh aspect ratios become large, which implies a severe element side-length discrepancy. In this work, an aspect ratio dependent lumped mass (ARLM) formulation is proposed for serendipity elements, i.e., the two-dimensional eight-node and three dimensional twenty-node quadratic elements for linear problems. In particular, a generalized parametric lumped mass matrix template taking into account the mesh aspect ratios is introduced to examine the frequency accuracy of serendipity elements. This generalized lumped mass matrix template completely meets the mass conservation and non-negativity requirements. Subsequently, analytical frequency error estimates are developed for serendipity elements, which clearly illustrate the relationship between the frequency accuracy and element aspect ratios. Accordingly, optimal mass parameters are obtained as the functions of element aspect ratios through solving a constrained optimization problem for frequency accuracy. It turns out that the resulting aspect ratio dependent lumped mass matrices yield much more accurate frequency solutions, in comparison to the diagonal scaling lumped mass (HRZ) matrices and the mid-node lumped mass (MNLM) matrices without consideration of the element aspect ratios, especially for finite element discretizations with severe element side-length discrepancy. The superior accuracy and robustness of the proposed ARLM over HRZ and MNLM are consistently demonstrated by numerical examples.

This paper presents an extension of the authors’ previously developed interface coupling technique for 2D problems to 3D problems. The method combines the strengths of the Discrete Element Method (DEM), known for its adeptness in capturing discontinuities and non-linearities at the microscale, and the Boundary Element Method (BEM), known for its efficiency in modelling wave propagation within infinite domains. The 3D formulation is based on spherical discrete elements and bilinear quadrilateral boundary elements. The innovative coupling methodology overcomes a critical limitation by enabling the representation of discontinuities within infinite domains, a pivotal development for large-scale dynamic problems. The paper systematically addresses challenges, with a focus on interface compatibility, showcasing the method’s accuracy through benchmark validation on a finite rod and infinite spherical cavity. Finally, a model of a column embedded into the ground illustrates the versatility of the approach in handling complex scenarios with multiple domains. This innovative coupling approach represents a significant leap in the integration of DEM and BEM for 3D problems and opens avenues for tackling complex and realistic problems in various scientific and engineering domains.

Abstract

This paper presents an approach to evaluate the failure of arbitrarily inclined interfaces using FE models with structured spatial discretization, providing accurate prediction of crack propagation along paths known a priori that are not constrained to the element boundaries. The combination of algorithms for the generation of structured discretization of representative polycrystalline microstructures with novel cohesive element formulations allow modelling the failure of complex topologies along rasterised boundaries, with noticeably higher computational efficiency and comparable accuracy. Two formulations of raster cohesive elements are presented, adopting either elastic-brittle or Tvergaard–Hutchinson traction separation laws. The formulations proposed are first validated comparing the failure of the interface within bi-crystal structures discretised using hexahedral elements either within a structured mesh (i.e. with rasterised boundaries) or an unstructured mesh (i.e. with planar boundary). Subsequently, the effectiveness of the formulations is demonstrated comparing the inter-granular crack propagation within complex polycrystalline microstructures. The behaviour of the novel cohesive element formulation in structured meshes consisting of regular hexahedral elements is in excellent agreement with the deformation and failure of classic cohesive element formulations placed along the planar boundaries of unstructured meshes consisting of tetrahedral elements. The higher computational cost of the raster cohesive elements is more than compensated by the increase in computational efficiency of structured meshes when compared to unstructured meshes, leading to a reduction of the simulation time of up to over 200 times for the simulations presented in the paper, thus allowing the simulation of large domains.

Time discontinuous Galerkin space-time finite element method (ST/FEM) can be used for developing arbitrary high-order accurate and unconditionally stable time integration schemes for elastodynamics problems. The existing ST/FEMs can be classified as the single-field and two-field ST/FEM: in the former method, either displacement or velocity, is independent and discontinuous in time. In contrast, in the latter method, both displacement and velocity fields are independent and discontinuous in time. Both methods have third-order accuracy for linear interpolation in time, higher than typical time integration schemes used in semi-discretized. However, these methods currently lack a unified computational framework, so each method requires a separate implementation. Therefore, the main goal of the present study is to develop a generalized computational framework that can facilitate the derivation and implementation of the existing linear-in-time ST/FEMs in a unified manner. This framework is developed by realizing that existing methods differ through the treatments of displacement-velocity relationships, which can be unified through displacement functions. In addition, by employing this framework, a new ST/FEM, which is designated as LC v-ST/FEM, is derived from the linear combination of displacement functions of single-field and two-field ST/FEMs. LC v-ST/FEM contains a user-defined parameter (alpha in [0,1]), which can be used for controlling the high-frequency dissipation characteristics. From finite difference analysis and numerical solutions of benchmark problems, it is demonstrated that the proposed method is the third order accurate in time, unconditionally stable, and contains negligible numerical dispersion error for all (0 le alpha le 1). Moreover, for (alpha ne 0), the method can attenuate the spurious high-frequency components from the velocity and displacement fields.

An alternative approach is proposed and applied to solve boundary value problems within the strain gradient elasticity theory. A mixed variation formulation of the finite element method (FEM) based on the concept of the Galerkin method is used. To construct finite-dimensional subspaces separate approximations of displacements, deformations, stresses, and their gradients are implemented by choosing the different sets of piecewise polynomial basis functions, interrelated by the stability condition of the mixed FEM approximation. This significantly simplifies the pre-requirement for approximating functions to belong to class C¹ and allows one to use the simplest triangular finite elements with a linear approximation of displacements under uniform or near-uniform triangulation conditions. Global unknowns in a discrete problem are nodal displacements, while the strains and stresses and their gradients are treated as local unknowns. The conditions of existence, uniqueness, and continuous dependence of the solution on the problem’s initial data are formulated for discrete equations of mixed FEM. These are solved by a modified iteration procedure, where the global stiffness matrix for classical elasticity problems is treated as a preconditioning matrix with fictitious elastic moduli. This avoids the need to form a global stiffness matrix for the problem of strain gradient elasticity since it is enough to calculate only the residual vector in the current approximation. A set of modeling plane crack problems is solved. The obtained solutions agree with the results available in the relevant literature. Good convergence is achieved by refining the mesh for all scale parameters. All three problems under study exhibit specific qualitative features characterizing strain gradient solutions namely crack stiffness increase with length scale parameter and cusp-like closure effect.

The present endeavour numerically exploits the use of a phase-field model to simulate and investigate fracture patterns, deformation mechanisms, damage, and mechanical responses in a human vertebra after the incision of pedicle screws under compressive regimes. Moreover, the proposed phase field framework can elucidate scenarios where different damage patterns, such as crack nucleation sites and crack trajectories, play a role after the spine fusion procedure, considering several simulated physiological movements of the vertebral body. Spatially heterogeneous elastic properties and phase field parameters have been computationally derived from bone density estimation. A convergence analysis has been conducted for the vertebra-screws model, considering several mesh refinements, which has demonstrated good agreement with the existing literature on this topic. Consequently, by assuming different angles for the insertion of the pedicle screws and taking into account a few vertebral motion loading regimes, a plethora of numerical results characterizing the damage occurring within the vertebral model has been derived. Overall, the phase field results confirm and enrich the current literature, shed light on the medical community, which will be useful in enhancing clinical interventions and reducing post-surgery bone failure and screw loosening. The proposed computational approach also investigates the effects in terms of fracture and mechanical behaviour of the vertebral-screws body within different metastatic lesions opening towards major life threatening scenarios.

The present study aims to develop an original solid-like shell element for large deformation analysis of hyperelastic shell structures in the context of isogeometric analysis (IGA). The presented model includes a new variable to describe the thickness change of the shell and allows for the application of unmodified three-dimensional constitutive laws defined in curvilinear coordinate systems and the analysis of variable thickness shells. In this way, the thickness locking affecting standard solid-shell-like models is cured by enhancing the thickness strain by exploiting a hierarchical approach, allowing linear transversal strains. Furthermore, a patch-wise reduced integration scheme is adopted for computational efficiency reasons and to annihilate shear and membrane locking. In addition, the Mixed-Integration Point (MIP) format is extended to hyperelastic materials to improve the convergence behaviour, hence the efficiency, in Newton iterations. Using benchmark problems, it is shown that the proposed model is reliable and resolves locking issues that were present in the previously published isogeometric solid-shell formulations.