Computational Mechanics最新文献

In this paper, we present a novel adaptive continuous–discontinuous approach for the analysis of phase field fracture. An initial trimmed hexahedral (TH) mesh is created by cutting a hexahedral background grid with the boundary of the solid domain. Octree-based adaptive mesh refinement is performed on the initial TH mesh based on an energy-based criterion to accurately resolve the damage evolution along the phase field crack. Critical damage isosurfaces of the phase field are used to convert fully developed phase field cracks into discontinuous discrete cracks. Mesh coarsening is also performed along the discontinuous discrete cracks to reduce the computational cost. Three-dimensional problems of quasi-brittle fracture are investigated to verify the effectiveness and efficiency of the present adaptive continuous–discontinuous approach for the analysis of phase field fracture.

Abstract

This is Part II of a multipart article on a hyperelastic extended Kirchhoff–Love shell model with out-of-plane normal stress. We introduce an isogeometric discretization method for incompressible materials and present test computations. Accounting for the out-of-plane normal stress distribution in the out-of-plane direction affects the accuracy in calculating the deformed-configuration out-of-plane position, and consequently the nonlinear response of the shell. The return is more than what we get from accounting for the out-of-plane deformation mapping. The traction acting on the shell can be specified on the upper and lower surfaces separately. With that, the model is now free from the “midsurface’ location in terms of specifying the traction. In dealing with incompressible materials, we start with an augmented formulation that includes the pressure as a Lagrange multiplier and then eliminate it by using the geometrical representation of the incompressibility constraint. The resulting model is an extended one, in the Kirchhoff–Love category in the degree-of-freedom count, and encompassing all other extensions in the isogeometric subcategory. We include ordered details as a recipe for making the implementation practical. The implementation has two components that will not be obvious but might be critical in boundary integration. The first one is related to the edge-surface moment created by the Kirchhoff–Love assumption. The second one is related to the pressure/traction integrations over all the surfaces of the finite-thickness geometry. The test computations are for dome-shaped inflation of a flat circular shell, rolling of a rectangular plate, pinching of a cylindrical shell, and uniform hydrostatic pressurization of the pinched cylindrical shell. We compute with neo-Hookean and Mooney–Rivlin material models. To understand the effect of the terms added in the extended model, we compare with models that exclude some of those terms.

In this work, we propose an efficient smoothed particle hydrodynamics (SPH) method for simulating laser powder bed fusion (LPBF). The multi-physics process of LPBF, including the heat transfer and phase change with complex boundaries, is accurately resolved by a novel heat source model and a modified continuous surface force based on a corrected surface delta function. Moreover, we also develop an efficient tensile instability control algorithm for preventing the pressure oscillations. The present method is implemented in a GPU-accelerated framework, and its performance is well demonstrated by simulating the LPBF processes with both single-layer and multi-layer powder beds (with the help of surface reconstruction). The numerical results are compared well with the experimental ones which clearly verify the ability of the present method in capturing the complex physical phenomenon of LPBF.

This study presents an innovative approach for developing a reduced-order model (ROM) tailored specifically for nearly incompressible materials at large deformations. The formulation relies on a three-field variational approach to capture the behavior of these materials. To construct the ROM, the full-scale model is initially solved using the finite element method (FEM), with snapshots of the displacement field being recorded and organized into a snapshot matrix. Subsequently, proper orthogonal decomposition is employed to extract dominant modes, forming a reduced basis for the ROM. Furthermore, we efficiently address the pressure and volumetric deformation fields by employing the k-means algorithm for clustering. A well-known three-field variational principle allows us to incorporate the clustered field variables into the ROM. To assess the performance of our proposed ROM, we conduct a comprehensive comparison of the ROM with and without clustering with the FEM solution. The results highlight the superiority of the ROM with pressure clustering, particularly when considering a limited number of modes, typically fewer than 10 displacement modes. Our findings are validated through two standard examples: one involving a block under compression and another featuring Cook’s membrane. In both cases, we achieve substantial improvements based on the three-field mixed approach. These compelling results underscore the effectiveness of our ROM approach, which accurately captures nearly incompressible material behavior while significantly reducing computational expenses.

This paper presents a novel framework combining proper generalized decomposition (PGD) with the shooting method to determine the steady-state response of nonlinear dynamical systems upon a general periodic input. The proposed PGD approximates the response as a low-rank separated representation of the spatial and temporal dimensions. The Galerkin projection is employed to formulate the subproblem for each dimension, then the fixed-point iteration is applied. The subproblem for the spatial vector can be regarded as computing a set of reduced-order basis vectors, and the shooting problem projected onto the subspace spanned by these basis vectors is defined to obtain the temporal coefficients. From this procedure, the proposed framework replaces the complex nonlinear time integration of the full-order model with the series of solving simple iterative subproblems. The proposed framework is validated through two descriptive numerical examples considering the conventional linear normal mode method for comparison. The results show that the proposed shooting method based on PGD can accurately capture nonlinear characteristics within 10 modes, whereas linear modes cannot easily approximate these behaviors. In terms of computational efficiency, the proposed method enables CPU time savings of about one order of magnitude compared with the conventional shooting methods.

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.