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The phase-field method is a very effective way to simulate arbitrary crack nucleation, propagation, bifurcation, and the formation of complex crack networks. The diffusion-based method is suitable for multi-field coupling fracture problems. In this paper, a parallel algorithm of the thermo-elastic coupled phase-field model is implemented in commercial finite element code Abaqus/Explicit. The algorithm is applied to simulate the dynamic and quasi-static brittle fracture of thermo-elastic materials. Further, it is adopted on a structured mesh combined with first-order explicit integrators. Several examples of the quasi-static and dynamic cases of single crack, as well as multi-crack initiation and propagation under thermal shock, are given to demonstrate the robustness of the algorithm. The source code and tutorials provide an effective way to simulate crack nucleation and propagation in multi-field coupling problems.

The phase field method is a very effective method to simulate arbitrary crack propagation, branching, convergence and complex crack networks. However, most of the current phase-field models mainly focus on tensile fracture problems, which is not suitable for rock-like materials subjected to compression and shear loads. In this paper, we derive the driving force of phase field evolution based on Mohr–Coulomb criterion for rock and other materials with shear frictional characteristics and develop a three-dimensional explicit parallel phase field model. In spatial integration, the standard finite element method is used to discretize the displacement field and the phase field. For the time update, the explicit central difference scheme and the forward difference scheme are used to discretize the displacement field and the phase field respectively. These time integration methods are implemented in parallel, which can tackle the problem of the low computational efficiency of the phase field method to a certain extent. Then, three typical benchmark examples of dynamic crack propagation and branching are given to verify the correctness and efficiency of the explicit phase field model. At last, the failure processes of rock-like materials under quasi-static compression load are studied. The simulation results can well capture the compression-shear failure mode of rock-like materials.

A coupled thermo-mechanical bond-based peridynamical (TM-BB-PD) method is developed to simulate thermal cracking processes in rocks. The coupled thermo-mechanical model consists of two parts. In the first part, temperature distribution of the system is modeled based on the heat conduction equation. In the second part, the mechanical deformation caused by temperature change is calculated to investigate thermal fracture problems. The multi-rate explicit time integration scheme is proposed to overcome the multi-scale time problem in coupled thermo-mechanical systems. Two benchmark examples, i.e., steady-state heat conduction and transient heat conduction with deformation problem, are performed to illustrate the correctness and accuracy of the proposed coupled numerical method in dealing with thermo-mechanical problems. Moreover, two kinds of numerical convergence for peridynamics, i.e., m - and δ -convergences, are tested. The thermal cracking behaviors in rocks are also investigated using the proposed coupled numerical method. The present numerical results are in good agreement with the previous numerical and experimental data. Effects of PD material point distributions and nonlocal ratios on thermal cracking patterns are also studied. It can be found from the numerical results that thermal crack growth paths do not increases with changes of PD material point spacing when the nonlocal ratio is larger than 4. The present numerical results also indicate that thermal crack growth paths are slightly affected by the arrangements of PD material points. Moreover, influences of thermal expansion coefficients and inhomogeneous properties on thermal cracking patterns are investigated, and the corresponding thermal fracture mechanism is analyzed in simulations. Finally, a LdB granite specimen with a borehole in the heated experiment is taken as an application example to examine applicability and usefulness of the proposed numerical method. Numerical results are in good agreement with the previous experimental and numerical results. Meanwhile, it can be found from the numerical results that the coupled TM-BB-PD has the capacity to capture phenomena of temperature jumps across cracks, which cannot be captured in the previous numerical simulations.

This paper strictly focuses upon novel designs of the time-integration algorithms as applied to structural dynamics systems with or without physical damping. The significant advances and contributions are summarized as follows: (1) the identity between the composite time-integration algorithms and the diagonally implicit Runge–Kutta family of algorithms are specifically established and demonstrated in order to clarify the originality, development, contribution, and pros/cons of the composite time-integration algorithms developed over the recent decades; (2) then, it is pointed out that the design of potential next-generation multi-stage time-integration algorithms with improved numerical properties can directly emanate from and already exist within the diagonally implicit Runge–Kutta–Nyström (DIRKN) computational framework itself, unlike composite-type time-integration methods paying efforts and attempting to design new algorithmic structures, although they are identical to and pertain primarily to the existing RK-type variants; (3) one- and two-stage DIRKN family of new algorithms and novel designs are taken into consideration for the first time, leading to novel sets of parameters with different numerical properties, which not only encompass existing methods by assigning two identical principal roots, but also produce new and novel designs by employing altogether distinctive principal roots; and finally, (4) the much coveted BN-stability feature and condition are additionally achieved and taken into consideration in order to optimize the design of parameters, which is competitive for nonlinear structural dynamics. Numerical examples are demonstrated to validate the analysis, new designs and the proposed overall efforts.

In this article, a simple way to determine algorithmic parameters included in time approximations of two-stage implicit time schemes is presented. To be specific, algorithmic parameters of time approximations are mathematically determined to give higher-order total energy convergence rates for conservative nonlinear problems while satisfying traditional linear accuracy requirements. Due to the use of newly proposed algorithmic parameters, two-stage implicit time schemes can possess enhanced total energy conserving capabilities for conservative nonlinear problems while providing improved linear performances when compared with those of the existing two-stage time schemes. Enhanced total energy conserving capabilities achieved through the use of newly proposed algorithmic parameters do not require any additional computational efforts when compared with the existing two-stage schemes. This article also explains that a certain standard type of two-stage implicit time schemes can reduce computational time and effort in linear analyses if effective coefficient matrices of the first and second stages are constructed identically. For the verification of improved numerical performances, linear and nonlinear benchmark problems are solved, and their numerical results are investigated to support the main discussions of this article.

In this study, a novel approach is proposed by integrating the finite element tearing and interconnecting (FETI) method into the B-differentiable equations (BDEs) method for the analysis of 3D elastic frictional contact problem with small deformations. The contact blocks are divided into several nonoverlapping substructures with nonconforming meshes on the contact surface and the interface between two adjacent substructures. The enforcement of contact conditions and interface continuity conditions is achieved by using dual Lagrange multipliers discretized on the slave surface, typically defined with fine meshes. The modified Boolean transformation matrix is utilized to convert the contact stress into the equivalent nodal force. For large-scale elastic contact problems, the equilibrium equations for substructures and the relationship between the relative displacements and contact stresses on the contact surfaces and interfaces (i.e., the contact flexibility matrix) are efficiently computed using the FETI method. Subsequently, the governing equations consisting of the contact equations, interface continuity equations, and equilibrium equations for each floating substructure are uniformly formulated as the BDEs. These BDEs can be solved using the B-differentiable damped Newton method (BDNM). The proposed method harnesses the parallel scalability of the FETI method and extends the applicability of the BDEs algorithm, benefiting from its ability to precisely satisfy the contact constraints and theoretically ensure convergence when solving large-scale contact problems. The Hilber/Hughes/Taylor (HHT) time integration scheme is employed to investigate elastic dynamic contact problems. Numerical examples demonstrate the accuracy, convergence rate, and parallel scalability of the proposed algorithm.

We propose two implicit numerical schemes for the low-rank time integration of stiff nonlinear partial differential equations. Our approach uses the preconditioned Riemannian trust-region method of Absil, Baker, and Gallivan, 2007. We demonstrate the efficiency of our method for solving the Allen–Cahn and the Fisher–KPP equations on the manifold of fixed-rank matrices. Our approach allows us to avoid the restriction on the time step typical of methods that use the fixed-point iteration to solve the inner nonlinear equations. Finally, we demonstrate the efficiency of the preconditioner on the same variational problems presented in Sutti and Vandereycken, 2021.

High-frequency (HF) radar plays a crucial role in the detection of far-range, stealth, and high-speed targets. Nevertheless, the echo signal of such targets typically exhibits a low signal-to-noise ratio (SNR) and significant amplitude fluctuations because their radar cross-section (RCS) accounting for the HF band is in the resonance region. While enhancing detection performance often requires long-time integration, existing algorithms inadequately consider the impact of amplitude fluctuation. In response to this challenge, this article introduces an improved approach based on coherent and non-coherent integration. Initially, coherent integration, employing the generalized Radon Fourier transform (GRFT), is utilized to derive a candidate detection set of targets’ range–time trajectories. This involves a joint solution for range migration (RM) and Doppler frequency migration (DFM) through a multi-parameter motion model search. Subsequently, the removal of low SNR pulses, followed by non-coherent integration, is implemented to mitigate amplitude fluctuation, referred to as Amplitude Fluctuation Suppression (AFS), and refine the detection outcomes. Both simulation and experiment results are provided to prove the effectiveness of the proposed AFS-GRFT algorithm.

The BeiDou navigation satellite system shows its potential for passive radar vessel target detection owing to its global-scale coverage. However, the restrained power budget from BeiDou satellite hampers the detection performance. To solve this limitation, this paper proposes a long-time optimized integration method to obtain an adequate signal-to-noise ratio (SNR). During the long observation time, the range migration, intricate Doppler migration, and noncoherence characteristic bring challenges to the integration processing. In this paper, first, the keystone transform is applied to correct the range walk. Then, considering the noncoherence of the entire echo, the hybrid integration strategy is adopted. To remove the Doppler migration and correct the residual range migration, the long-time integration is modeled as an optimization problem. Finally, the particle swarm optimization (PSO) algorithm is applied to solve the optimization problem, after which the target echo over the long observation time is well concentrated, providing a reliable detection performance for the BeiDou-based passive radar. Its effectiveness is shown by the simulated and experimental results.

In this work a poweful approach is presented to solve the time-fractional gas dynamics equation. In fact, we use a fictitious time variable y to convert the dependent variable w(x, t) into a new one with one more dimension. Then by taking a initial guess and implementing the group preserving scheme we solve the problem. Finally four examples are solved to illustrate the power of the offered method. nema