Implicit Algorithm Research Articles

An exact implementation of the generalized Zhang-Zhu criterion for elasto-plastic finite element calculations

The generalized Zhang-Zhu (GZZ) strength criterion was proposed as an extension to the Hoek-Brown criterion and the Mogi criterion. The introduction to mean normal stress results in a non-smooth and non-convex yield surface, which presents a challenge for updating plastic stress. Current research primarily focuses on modified smooth GZZ criteria or approximate solutions, which inevitably lead to increased computational costs or inaccuracies. In this paper, an accurate stress updating algorithm is proposed based on the original GZZ criterion. The algorithm operates entirely in the principal stress space, where numerical singularities at the intersection of yield surfaces are avoided by defining four different types of stress updating. This approach simplifies the GZZ criterion compared to its formulation in general stress space. The return mapping is employed to compute the updated stress and consistent stiffness matrix, facilitating calculations using both finite element implicit and explicit algorithms. Finally, the accuracy of the proposed method is validated using rock true triaxial test data and semi-analytical solutions for stresses and displacement around a circular opening under the GZZ criterion.

Optimization of Forming Parameters of Ti-3Al-2.5V Titanium Tubes by Differential Heating Push-Bending Based on a Modified Johnson−Cook Constitutive Model

This study addresses the challenges associated with the difficult formation and low quality of formed Ti-3Al-2.5V titanium alloy thin-walled tubes with small bending radii (r / D < 1.5, where r is the bending radius, D is the outer diameter of the tube) by proposing a differential heating push-bending (DHP-B) forming approach. Initially, the stress−strain curves of Ti-3Al-2.5V at various strain rates and temperatures were obtained through high-temperature uniaxial tensile tests. Subsequently, a modified Johnson−Cook (JC) model for Ti-3Al-2.5V within the temperature range of 25℃ to 800℃ was developed to enhance the accuracy of the finite element model. Based on the static implicit and dynamic explicit algorithms, a thermomechanically coupled finite element model for differential heating push-bending of titanium alloy thin-walled tubes has been established. Taking the heating temperature, friction coefficient, counterthrust force and rubber block thickness as the optimization parameters, the prediction model of the maximum wall thinning rate and thickening rate is obtained by using the response surface method, and the best parameter combination is obtained via the NSGA-II algorithm and the minimum distance selection method, which realizes the optimization of differential heating push-bending parameters, as verified by experiments in which the maximum wall-thinning rate is reduced to 9%, and the maximum wall thickness thickening rate is reduced to 14.2%, which improves the forming quality of the titanium tube.

Numerical simulation on multi-well fracturing considering multiple thin layers in vertical direction

Multiwell fracturing is a key technology for developing shale gas and shale oil reservoirs. In this study, a multiple planar 3D (PL3D) fracture simulator that can capture multiple thin layers was developed to examine the propagation of multiple fractures during multicluster fracturing in multiple horizontal wells. The simulator considers multiple thin layers in the vertical direction. The results of the model are validated against the analytical solution of a single radial fracture and the implicit level set algorithm (ILSA). Using the simulator, a series of numerical simulations based on the field case are performed to investigate the fracture propagation mechanism of multiwell fracturing. The completion sequence, well placement pattern, well spacing, and cluster spacing are investigated to optimize the treatment parameters. The effective fracture area is used to quantitatively describe the stimulation effect. The adaptability of the completion sequence and well placement pattern is also analysed from the perspective of “frac hits”. The results show that the completion sequence has a critical influence on the stimulation effect and fracture geometry. From the perspective of avoiding “frac-hit” fractures, fracturing the low-stress layer can form an “artificial stress barrier”, which slightly protects the well from interference from other fractures. The staggered well pattern is better than the stacked well pattern. Compared with the stacked pattern, the staggered pattern can reduce the overlap area of fractures by 80 %, which greatly reduces the probability of “frac-hits”. With increasing well spacing from 200 m to 500 m, the fracture area increases by 25 %, and the degree of uneven stimulation between the two pay zones also increases by 6 %. Considering that a small well spacing is prone to “frac hits”, a large well spacing leads to an unstimulated area between two wells, and a 350 m well spacing is optimal. The effective fracture area decreases slightly with increasing perforation cluster spacing, but the fracture geometry becomes much more regular. The results can be helpful for the field design of multiwell fracturing.

A Projection-Type Implicit Algorithm for Finding a Common Solution for Fixed Point Problems and Variational Inequality Problems

This paper deals with the problem of finding a common solution for a fixed point problem for strictly pseudocontractive mappings and for a certain variational inequality problem. We propose a projection-type implicit averaged algorithm and establish the strong convergence of the sequences generated by this method to the common solution for the fixed point problem and the variational inequality problem. In order to illustrate the feasibility of the hypotheses and the superiority of our theoretical results over the existing literature, an example is also presented.

Magneto-viscoelastic rod model for hard-magnetic soft rods under 3D large deformation: Theory and numerical implementation

The main purpose of this work is to develop a three-dimensional (3D) viscoelastic rod model for hard-magnetic soft (HMS) rods under large deformation which are widely used active structures in soft robotics. To do so, the Simo’s viscoelasticity theory has been rationally incorporated into the geometrically exact 3D curved rod model. The proposed model includes the deformation modes of axial tension, shear, bending, and torsion, which is applicable to the HMS rods with arbitrarily initial curved and twisted geometries under 3D large deformation. The viscoelastic constitutive equations of the HMS rod in the present formulation are formulated, which include the general relaxation functions. To obtain the expression for the magnetic load, the rotation-based magnetic free energy density is introduced, and the governing equations of the HMS rod with magnetic load and body force are presented. To obtain the numerical implementation, an implicit time integration algorithm that simply extends the generalized-α method for the rotation group, and the corresponding variational formulation and its linearization of the rod model are derived. To validate the model, five numerical examples, including 2D dynamic buckling, 3D static, and 3D dynamic problem are presented. The dynamic problems include the dynamic snap-through behavior of a bistable HMS arch and damped oscillation of a quarter arc cantilever under 3D deformation. The simulation results show good agreement with the results reported in the literature.

A novel semi-explicit numerical algorithm for efficient 3D phase field modelling of quasi-brittle fracture

Phase field models have become an effective tool for predicting complex crack configurations including initiation, propagation, branching, intersecting and merging. However, several computational issues have hindered their utilisation in engineering practice, such as the convergence challenge in implicit algorithms, numerical stability issues in explicit methods and significant computational costs. Aiming to providing a more efficient numerical algorithm, this work integrates the explicit integral operator with the recently developed neighbored element method, for the first time, to solve the coupled governing equations in phase field models. In addition, the damage irreversibility can be ensured automatically, avoiding the need to introduce extra history variable for the maximum driving force in traditional algorithms. Six representative fracture benchmarks with different failure modes are simulated to verify the effectiveness of the proposed method, including the multiple cracks in heterogeneous concrete at mesoscale. It is found that this semi-explicit numerical algorithm yields consistent crack profiles and load capacities for all examples to the available experimental data and literature. In particular, the computational cost is significantly reduced when compared to the traditional explicit modelling. Therefore, the presented numerical algorithm is highly attractive and promising for phase-field simulations of complicated 3D solid fractures in structural-level engineering practices.

Three-dimensional numerical simulation and experimental validation of airborne ground-penetrating radar based on CNCS-FDTD method under undulating terrain conditions

Traditional Finite Difference Time Domain (FDTD) approaches face challenges with increased computational demands and errors as terrain complexity and flight altitude rising. This study introduces the Crank-Nicolson-Cycle-Sweep-FDTD (CNCS-FDTD) method, enhancing airborne ground penetrating radar (GPR) simulations over undulating terrains. CNCS-FDTD, as an unconditionally stable implicit algorithm, overcomes these by allowing larger time steps without the constraints of the Courant-Friedrichs-Lewy (CFL) condition. Our research aims to assess how CNCS-FDTD can improve computational efficiency and accuracy in modeling airborne GPR responses across varied terrains. Initial simulations using a three-dimensional graben model indicate that CNCS-FDTD can maintain calculation stability with significantly larger time steps, reducing computational time by 42%. To further verify the reliability of the numerical simulation results, the paper also presents experimental tests of the undulating terrain model. By comparing the numerical and physical simulation results of models under different flight altitudes and terrain conditions, the accuracy of the simulation results is validated.

Implicit-integral dynamic optimization based on spatial partitioning and temporal segmentation for the power jumps of renewable energy sources

With the widespread development of renewable energy sources, the increasing short-term power volatility poses a significant challenge to the normal operation of power systems. This paper presents a two-timescale adaptive dynamic optimization (TADO) strategy based on implicit trapezoidal integral to address dynamic state fluctuations due to renewable power jumps. Firstly, a static optimization model is established utilizing the model predictive control (MPC) to obtain the optimal states on a longer temporal scale, considering the uncertain renewable power jumps. Next, provide a partitioning method based on modularity to divide the power system into several independent subsystems with constant tie-line power. Using an implicit trapezoidal integral algorithm, each subsystem's dynamic reactive power optimization model is constructed with an appropriate discrete time granularity, accounting for the electromechanical transient processes due to renewable power jumps. Then, to improve the solving efficiency of the dynamic optimization model, a new solution method is proposed based on temporal segmentation. The whole process's optimal state trajectory is achieved by splicing all sub-optimization results. Finally, case studies conducted on the IEEE 9-bus power system and IEEE 39-bus power system validate the feasibility of the proposed strategy.

A fully implicit, asymptotic-preserving, semi-Lagrangian algorithm for the time dependent anisotropic heat transport equation

In this paper, we extend the operator-split asymptotic-preserving, semi-Lagrangian algorithm for time dependent anisotropic heat transport equation proposed in Chacón et al. (2014) [18] to use a fully implicit time integration with backward differentiation formulas. The proposed implicit method can deal with arbitrary heat-transport anisotropy ratios χ∥/χ⊥⋙1 (with χ∥, χ⊥ the parallel and perpendicular heat diffusivities, respectively) in complicated magnetic field topologies in an accurate and efficient manner. The implicit algorithm is second-order accurate temporally and demonstrates an accurate treatment at boundary layers (e.g., island separatrices), which was not ensured by the operator-split implementation. The condition number of the resulting algebraic system is independent of the anisotropy ratio, and is inverted with preconditioned GMRES. We propose a simple preconditioner that renders the finite-dimensional linear operator compact, resulting in mesh-independent convergence rates for topologically simple magnetic fields, and convergence rates scaling as ∼(NΔt)1/4 (with N the total mesh size and Δt the timestep) in topologically complex magnetic-field configurations. We demonstrate the accuracy and performance of the approach with test problems of varying complexity, including an analytically tractable boundary-layer problem in a straight magnetic field, and a topologically complex magnetic field featuring magnetic islands with extreme anisotropy ratios (χ∥/χ⊥=1010).

Fracture modeling of curved composite shell structures using augmented finite element method

This paper presents an augmented finite element method for the fracture modeling of curved composite shell structures considering geometric nonlinear effects. We first derive the composite shell AFEM (CS-AFEM) formulation using a dynamic implicit algorithm, which enhances the numerical convergence when dealing with unstable crack propagations. Besides, the cohesive zone model featuring with shell kinematics is incorporated into the CS-AFEM to describe the quasi-brittle fracture behaviors of composite laminates. Furthermore, a coordinate transformation scheme is proposed to describe both the local material orientation and the crack evolving path in the curved shell structures. Finally, several benchmark examples are numerically investigated, and the capability of the proposed method in modeling the arbitrary evolving cracks in curved composite shell structures has been thoroughly demonstrated.

An Efficient Iterative Method for Vehicle-Track Nonlinear Coupled Dynamic Analysis Based on Explicit and Implicit Algorithms

An Efficient Iterative Method for Vehicle-Track Nonlinear Coupled Dynamic Analysis Based on Explicit and Implicit Algorithms

The modified Mohr-Coulomb model considering softening effect and intermediate principal stress

Based on the Mohr-Coulomb model, a modified Mohr-Coulomb model that can consider both softening effect and intermediate principal stress effect was formed by introducing plastic internal variables that characterize material damage and a modified stress Rhodes angle. The modified model is numerically integrated by implicit algorithm, the physical meaning of the model parameters of the modified model is clarified, and the multivariate analysis and sensitivity analysis of the model parameters are carried out. Combined with the results of the gypsum triaxial test, the model parameters of the modified model were determined by the equation solving and RBF neural network optimization method, and the validity of the model was verified. The research results showed that: The modified model established can consider both the material softening effect and the influence of the intermediate principal stress. Compared with the traditional Mohr-Coulomb model, the calculation results of the modified model are more consistent with the test results qualitatively and quantitatively. The comprehensive use of direct determination, equation solving and RBF neural network optimization methods can effectively determine the model parameters of the modified model, and the determination process is relatively simple. It is suggested that this combined method can be used for subsequent determination of more complex constitutive model parameters. Compared with the traditional Mohr-Coulomb model, the modified model only needs one more true-triaxial test to determine the model parameters. In addition, the experience of using the parameters of the traditional Mohr-Coulomb model in a large number of engineering practices can be used to improve the model, so the modified model has great potential for engineering applications.

The effects of energy accommodation of reflected gas molecules on flow structures during expired spacecraft reentry

To study the influence of energy accommodation of scattering gas molecules on flow fields during large expired spacecraft reentry, a more elaborated gas-surface interaction model, compared with full Maxwellian diffuse model, is employed in implicit algorithms based on Boltzmann model equation. The characteristic distributions around cylinder at different fluid regimes are accordingly obtained by implicit algorithms, Navier-Stokes solver and DSMC ((Direct Simulation Monte Carlo) method. And the consistency of these results is verified. It is confirmed that present algorithms are capable of solving external flow problems covering various fluid regimes. Then the simulation results see that under current conditions set in the paper, pressure and temperature are proportional to wall activation (ω=Tw/T∞, Tw is surface temperature, T∞ denotes as free stream temperature), but their amplitudes alter with ω at different fluid regimes. As for the effects of energy accommodation coefficients (αe), both pressure and temperature profiles vary in a linear way with αe. However, the variation ranges of these parameters are diverse with regard to different fluid regimes. These observations are favor to the construction of efficient forecasting software, which could predict the flight path of large defunct spacecraft. In this forecasting software, the external ballistics computations and aerothermodynamic simulations are synchronously carried out.

Fluid–Structure Interaction Analysis of Manta-Bots with Self-Induced Vertical Undulations during Fin-Based Locomotion

Driven by the demands of ocean exploration, an increasing number of manta ray-inspired robots have been designed and manufactured, primarily utilizing flexible skeletons combined with motor-driven mechanisms. However, the mechanical analysis of these designs remains underdeveloped, often relying on simplistic imitation of biological prototypes and typically neglecting the vertical motion induced by pectoral fin flapping. This paper presents a fluid–structure interaction analysis framework that couples rigid body motion with elastic deformation using flexible multibody dynamics and the vortex particle method. An implicit iterative algorithm with Aitken relaxation is employed to address added-mass instability, and the framework has been validated against experimental data. An analysis of a representative manta-bot model shows that self-induced vertical undulations reduce the thrust coefficient by approximately 40% compared to fixed vertical degrees of freedom, while slightly improving overall propulsive efficiency. The study also highlights the critical role of mass distribution in manta-bots, noting that excessive focus on complex pectoral fin movements and large fin mass can significantly reduce thrust by increasing vertical displacement, ultimately proving counterproductive.

A reduced-order model of thermo-viscoelastic filaments in a material extrusion process

High-resolution and high-efficiency simulations are a great challenge in a material extrusion process. This paper developed a reduced-order thermo-viscoelastic model based on discrete differential geometry. This model successfully simulated the dynamic behaviors, heat transfer, and deformations of filaments during the material extrusion with high computational efficiency. The kinematics of filaments was constructed using a centerline and time-parallel frame combined scheme, and the dynamic system was derived based on the elastic and viscous potential energies of stretching, bending, and twisting. A Newmark implicit algorithm was applied to solve the dynamic equations of the model, and a collision detection and response strategy for filaments was proposed. The convergence, accuracy, and efficiency of this model were validated and tested from various aspects. This model was then utilized to examine the impact of printing parameters on the temperature distribution and sagging deformation of an overhanging filament, and its extended applications were further demonstrated.

Quasi-Newton positivity-preserving scheme for the Spalart-Allmaras turbulence model using unstructured grids

A new implicit method for solving the Spalart-Allmaras (SA) turbulence model is proposed using a formal second-order upwind scheme on unstructured grids for steady-state flows. The mean-flow set of equations and the SA model equation are solved in a loosely coupled manner. While the Jacobian of the mean-flow equations is derived from a low-order approximation of the residual (i.e. the defect-correction approach), the SA Jacobian is obtained by modifying a direct linearization of the second-order residual. The modified Jacobian is designed to form an M-matrix without the artificial time derivative commonly used for steady-state problems. Namely, no time step is involved in solving the SA model equation. Thanks to a special design of the M-matrix, the positivity of the SA model working variable and the convergence of the SA model linearized problem (fixed-point iterative problem) is guaranteed. To further enhance the overall stability of the flow solver, the Runge-Kutta implicit algorithm is employed, allowing CFL numbers as high as one to two thousand for the mean-flow equations. A second-order upwind scheme is achieved with a least-square method using a limiter to suppress oscillation in the solution. However, second-order upwind scheme discretization of turbulence models may lack robustness, especially when using unstructured grids. The limiter can hinder the convergence of the non-linear iterations, especially of the turbulence model. A square root transformation is applied to the dependent variable of a baseline SA model to improve the non-linear convergence characteristics. Three variants of the SA model are implemented and tested together with two limiters. Numerical simulations of a broad range of aerodynamic applications are conducted. The transformed variant exhibits the least sensitivity to the limiter type.

IRDA: Implicit data augmentation for deep imbalanced regression

Imbalanced data distributions are prevalent in real-world classification and regression tasks. Data augmentation is a commonly employed technique to mitigate this issue, with implicit methods gaining attention for their effectiveness and efficiency. However, implicit data augmentation methods have not been extensively explored in the context of regression tasks. To address this gap, we introduce IRDA, a novel learning method for regression that incorporates implicit data augmentation. Our approach includes developing a new augmentation strategy specifically tailored for deep imbalanced regression tasks, and a regression loss function that is suitable for real-world data with imbalanced label distributions and non-uniformly distributed features. We derive an easily computable surrogate loss and propose two implicit data augmentation algorithms, one incorporating meta-learning and one without. Additionally, we provide a regularization perspective to offer a deeper understanding of IRDA. We evaluate IRDA on five datasets, including a large-scale dataset, demonstrating its effectiveness in mitigating the adverse effects of imbalanced data distribution and its adaptability to various regression tasks.

A modified J model for efficiently calculating the electromagnetic fields of ReBCO no-insulation pancake coils using an explicit–implicit hybrid algorithm

Rare-earth (Re)Ba2Cu3O7−x (ReBCO) no-insulation (NI) coil is widely concerned due to its excellent electromagnetic and thermal properties. However, the presence of the turn-to-turn shunts in NI coils leads to that complexity of numerical simulation is increased. In this paper, a modified J model is proposed and the corresponding explicit–implicit hybrid algorithm is designed to calculate NI coils. The numerical results are in good agreement with the experimental data and the circuit model. The homogenization model is also proposed to simulate the large-scale NI coils in the background magnets. The modified J model has good accuracy and fast calculation speed, which can also be used to solve electromagnetic fields of insulation coils efficiently.

Modeling confined concrete behavior in finite element with a non-orthogonal elastoplastic model

The accuracy of a constitutive model for confined concrete largely relies on its capability to capture concrete's dilatancy behavior. In this paper, a non-orthogonal flow rule (NFR) is used to reasonably characterize the concrete's volume change law in relation to the stress state without the necessity for a plastic potential function. Then, the non-orthogonal plastic model is implemented in the finite element (FE) software ABAQUS using the implicit stress update algorithm, which employs the line search method and the numerical consistent tangent stiffness matrix to ensure the robustness and computational efficiency of FE analysis. Finally, FE simulations are performed to evaluate the constitutive model's capabilities in actively and passively confined concrete. In the latter case, steel tubes restrict the concrete's lateral deformation. An analysis and discussion are conducted regarding the impact of dilatancy behavior on concrete-filled steel tube (CFST) columns. The consistency between experimental data and simulation results demonstrates that the FE modeling with a non-orthogonal constitutive model provides an effective tool to describe the behavior of confined concrete.

A Novel Approach to Linear and Nonlinear Time-History Analysis of Structures: Gauss–Lobatto–Hermite 4-Point (GLH-4P) Method

An efficient time integration scheme has been devised for conducting time-history dynamic analysis of single-degree-of-freedom (SDOF) structural systems. The developed scheme belongs to Runge-Kuta family. The primary innovation of the new formulation lies in the constructive cooperation between the Hermite interpolation and the Gauss–Lobatto integration, linked to address the second-order ordinary differential equation of motion (DEOM). To put the idea into practice, several Hermite interpolators are generated to exploit the information of the internal points of Gauss-Lobatto integrator. The 4-point Gauss–Lobatto formula is employed to numerically integrate acceleration and velocity functions. This novel methodology has been designated the Gauss–Lobatto–Hermite 4-point (GLH-4P) method. GLH-4P furnishes a generally applicable implicit algorithm for conducting both linear and nonlinear analyses of structures subjected to seismic excitation. The advantage of the proposed algorithm over the conventional techniques is its higher accuracy level. Besides, it offers better stability compared with the conventional methods such as the Newmark-β and Wilson-θ methods. Unlike the other methods, GLH-4P can handle high-frequency systems without reducing the step size. Numerical examples reveal the efficacy of the GLH-4P method when juxtaposed against existing methodologies.