Implicit Integration Research Articles

Smart modeling of soil-foundation interaction using coupled mechanisms: a numerical framework for liquefaction risk mitigation

This study investigates the impact of nearby structures on the cyclic settlement mechanisms of shallow foundations in liquefiable soils using a numerical model based on Biot’s porous media theory. The model predicts excess pore water pressure and settlement by coupling equilibrium and continuity equations, solved using an implicit time integration scheme. Soil nonlinearity under cyclic loading is represented using generalized plasticity, boundary surfaces, and non-associated models. Three scenarios are simulated to study the effect of spacing between light and heavy foundations and variation in acceleration intensity. Results show that as spacing between foundations increases, lateral displacement and settlement decrease. Excess pore water pressure generation also decreases with increased foundation spacing. Soil just below the foundation exhibits maximum settlement, decreasing with depth. When input acceleration increases from 0.1 g to 0.15 g and 0.2 g, settlement increases by 40%–55% and 90%–110% respectively for both light and heavy foundations, regardless of spacing. Excess pore water pressure also increases sharply with higher acceleration intensity. The findings highlight the importance of considering foundation-soil-foundation interaction effects in liquefaction-prone urban settings and provide insights for designing resilient shallow foundations. The advanced numerical modeling approach offers engineers a more informed way to mitigate liquefaction risk and build safer, more durable structures in earthquake-prone areas.

The Body Knows Better: Sensorimotor signals reveal the interplay between implicit and explicit Sense of Agency in the human mind

Sense of Agency (SoA) is the feeling of control over our actions. SoA has been suggested to arise from both implicit sensorimotor integration as well as higher-level decision processes. SoA is typically measured by collecting participants' subjective judgments, conflating both implicit and explicit processing. Consequently, the interplay between implicit sensorimotor processing and explicit agency judgments is not well understood. Here, we evaluated in one exploratory and one preregistered experiment (N = 60), using a machine learning approach, the relation between a well-known mechanism of implicit sensorimotor adaptation and explicit SoA judgments. Specifically, we examined whether subjective judgments of SoA and sensorimotor conflicts could be inferred from hand kinematics in a sensorimotor task using a virtual hand (VH). In both experiments participants performed a hand movement and viewed a virtual hand making a movement that could either be synchronous with their action or include a parametric temporal delay. After each movement, participants judged whether their actual movement was congruent with the movement they observed. Our results demonstrated that sensorimotor conflicts could be inferred from implicit motor kinematics on a trial by trial basis. Moreover, detection of sensorimotor conflicts from machine learning models of kinematic data provided more accurate classification of sensorimotor congruence than participants' explicit judgments. These results were replicated in a second, preregistered, experiment. These findings show evidence of diverging implicit and explicit processing for SoA and suggest that the brain holds high-quality information on sensorimotor conflicts that is not fully utilized in the inference of conscious agency.

A thermodynamically‐based fractional model combined viscoelastic‐viscoplastic‐ductile damage with application to fiber‐reinforced polymer composites

AbstractA thermodynamically‐based fractional viscoelastic‐viscoplastic‐damage constitutive model combined with continuous damage mechanics (CDM) theory was established, in order to describe the rate‐dependent nonlinear behavior of fiber‐reinforced polymer composites (FRPCs). The fractional Helmholtz free energy consists of four contributions: viscoelastic (VE), viscoplastic (VP), hardening and damage, in which the VE and VP parts are constructed by fractional Zener and Scott‐Blair (SB) element forms respectively. The constitutive equation is obtained through Helmholtz free energy for the fractional Zener model, and plastic flow and hardening evolution law are all derived in the process. The ductile damage, coupled to both VE and VP free energy parts, is introduced through fractional damage energy release rates to model the degradation of material properties. The corresponding strain energy release rate and dissipation contributions are also derived. The fractional implicit time integration algorithms of proposed model are presented. The model is applied to validate tests of FRPCs under various loading conditions. The model validation and comparison are presented by simulating experimental data and existing models in the literature. And the corresponding evolution of dissipated energy is discussed to further valid the characterization ability of the model.Highlights A thermodynamical fractional constitutive model was developed for FRPCs. The Helmholtz free‐energy potential for fractional Zener model is adopted. The physical significance of fractional order parameters is explored. Fractional implicit integration algorithm of proposed model is implemented. The validation and comparison of the model are presented under various loads.

A plastic correction algorithm for full-field elasto-plastic finite element simulations: critical assessment of predictive capabilities and improvement by machine learning

AbstractThis paper introduces a new local plastic correction algorithm that is aimed at accelerating elasto-plastic finite element (FE) simulations for structural problems exhibiting localised plasticity (around e.g. notches, geometrical defects). The proposed method belongs to the category of generalised multi-axial Neuber-type methods, which process the results of an elastic prediction point-wise in order to calculate an approximation of the full elasto-plastic solution. The proposed algorithm relies on a rule of local proportionality, which, in the context of J2 plasticity, allows us to express the plastic correction problem in terms of the amplitude of the full mechanical tensors only. This lightweight correction problem can be solved for numerically using a fully implicit time integrator that shares similarities with the radial return algorithm. The numerical capabilities of the proposed algorithm are demonstrated for a notched structure and a specimen containing a distribution of spherical pores, subjected to monotonic and cyclic loading. As a second point of innovation, we show that the proposed local plastic correction algorithm can be further accelerated by employing a simple meta-modelling strategy, with virtually no added errors. At last, we develop and investigate the merits of a deep-learning-based corrective layer designed to reduce the approximation error of the plastic corrector. A convolutional architecture is used to analyse the neighbourhoods of material points and outputs a scalar correction to the point-wise Neuber-type predictions. This optional brick of the proposed plastic correction methodology relies on the availability of a set of full elasto-plastic finite element solutions to be used as a training data-set.

Spatially mixed implicit–explicit schemes in hydro-mechanically coupled soil dynamics

Mixed implicit–explicit (IMEX) schemes combine elements of both implicit and explicit time integration methods. This allows to efficiently handle boundary value problems with severe differences in stiffness, such as often encountered in soil dynamics. A spatially mixed IMEX scheme, in which different parts of the computational domain may be treated implicitly or explicitly depending on their behaviour, is discussed in this work. Domain decomposition by means of a mortar contact discretisation is used for this purpose and multiple systems of equations are solved. The merits of the scheme in solving problems of soil dynamics are demonstrated, for which the governing equations are extended to hydro-mechanical coupling.

A novel iterative algorithm for nonlinear inelastic dynamic analysis of truss structures; MCB-Spline+Sp time integration method

The focus of this work is on the development of a novel algorithm for the nonlinear dynamic analysis of structures. In the sake of a general and easy-to-implement solution method, the tangential stiffness matrix is calculated by combining the Cauchy-Riemann equations derived from the differentiation of complex functions and using the nodal displacement perturbation vector. An implicit direct time integration method based on a cubic B-Spline is integrated with the incremental-iterative method using a quadrature rule based on the interpolation of spline functions, which is referred to as the MCB-Spline+Sp time integration method. A step-by-step algorithm for developing a computer code to implement this method is provided. The effectiveness of the proposed approach is evaluated through several nonlinear time history analyses, which demonstrate its accuracy and efficiency by observing less computational efforts and fast convergence rate. Moreover, a comparison is made between the developed method and well-established techniques such as the Newmark and Standard-Bathe time integration methods in combination with the iterative Newton-Raphson method.

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 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).

One-sweep moment-based semi-implicit-explicit integration for gray thermal radiation transport

Thermal radiation transport (TRT) is a time dependent, high dimensional partial integro-differential equation. In practical applications such as inertial confinement fusion, TRT is coupled to other physics such as hydrodynamics, plasmas, etc., and the timescales one is interested in capturing are often much slower than the radiation timescale. As a result, TRT is treated implicitly, and due to its stiffness and high dimensionality, is often a dominant computational cost in multiphysics simulations. Here we develop a new approach for implicit-explicit (IMEX) integration of gray TRT in the deterministic SN setting, which requires only one sweep per stage, with the simplest first-order method requiring only one sweep per time step. The partitioning of equations is done via a moment-based high-order low-order formulation of TRT, where the streaming operator and first two moments are used to capture the asymptotic stiff regimes of the streaming limit and diffusion limit. Absorption-reemission is treated explicitly, and although stiff, is sufficiently damped by the implicit solve that we achieve stable accurate time integration without incorporating the coupling of the high order and low order equations implicitly. Due to nonlinear coupling of the high-order and low-order equations through temperature-dependent opacities, to facilitate IMEX partitioning and higher-order methods, we use a semi-implicit integration approach amenable to nonlinear partitions. Results are demonstrated on thick Marshak and crooked pipe benchmark problems, demonstrating orders of magnitude improvement in accuracy and wallclock compared with the standard first-order implicit integration typically used.

Numerical model for time-dependent cracking and deflection of large-span concrete bridge structures

A new computational framework is proposed to analyse the simultaneous occurrence of multiple physical processes affecting the long-term behavior of large-scale concrete structures. Relying on the additivity of small strain, the framework comprehensively takes the nonlinearity of concrete creep, cyclic creep induced by traffic load, concrete shrinkage, concrete cracking, normal-strength steel rebars and prestressing strands relaxation into account, and other physical processes are further imagined. The algorithm is implemented with Abaqus/Standard with the aid of several user-supplied subroutines. The implicit time integration scheme is deemed appropriate for large-scale structures. The framework is systematically validated by several classical experimental results. Finally, a three-dimensional study of a three-span continuous rigid frame bridge with 270 m main span is performed. Comparison between the experimental and numerical results further reveals the contributions of various potential influencing factors. The proposed model shows promise for predicting the time-dependent performance of large-scale reinforcement concrete (RC) structures.

Implicit-explicit Runge-Kutta for radiation hydrodynamics I: Gray diffusion

Radiation hydrodynamics are a challenging multiscale and multiphysics set of equations. To capture the relevant physics of interest, one typically must time step on the hydrodynamics timescale, making explicit integration the obvious choice. On the other hand, the coupled radiation equations have a scaling such that implicit integration is effectively necessary in non-relativistic regimes. A first-order Lie-Trotter-like operator split is the most common time integration scheme used in practice, alternating between an explicit hydrodynamics step and an implicit radiation solve and energy deposition step. However, such a scheme is limited to first-order accuracy, and nonlinear coupling between the radiation and hydrodynamics equations makes a more general additive partitioning of the equations non-trivial. Here, we develop a new formulation and partitioning of radiation hydrodynamics with gray diffusion that allows us to apply (linearly) implicit-explicit Runge-Kutta time integration schemes. We prove conservation of total energy in the new framework, and demonstrate 2nd-order convergence in time on multiple radiative shock problems, achieving error 3–5 orders of magnitude smaller than the first-order Lie-Trotter operator split at the hydrodynamic CFL, even when Lie-Trotter applies a 3rd-order TVD Runge-Kutta scheme to the hydrodynamics equations.

Krylov-based adaptive-rank implicit time integrators for stiff problems with application to nonlinear Fokker-Planck kinetic models

We propose a high order adaptive-rank implicit integrators for stiff time-dependent PDEs, leveraging extended Krylov subspaces to efficiently and adaptively populate low-rank solution bases. This allows for the accurate representation of solutions with significantly reduced computational costs. We further introduce an efficient mechanism for residual evaluation and an adaptive rank-seeking strategy that optimizes low-rank settings based on a comparison between the residual size and the local truncation errors of the time-stepping discretization. We demonstrate our approach with the challenging Lenard-Bernstein Fokker-Planck (LBFP) nonlinear equation, which describes collisional processes in a fully ionized plasma. The preservation of the equilibrium state is achieved through the Chang-Cooper discretization, and strict conservation of mass, momentum and energy via a Locally Macroscopic Conservative (LoMaC) procedure. The development of implicit adaptive-rank integrators, demonstrated here up to third-order temporal accuracy via diagonally implicit Runge-Kutta schemes, showcases superior performance in terms of accuracy, computational efficiency, equilibrium preservation, and conservation of macroscopic moments. This study offers a starting point for developing scalable, efficient, and accurate methods for high-dimensional time-dependent problems.

Nonlinear dynamic analysis of shear- and torsion-free rods using isogeometric discretization and outlier removal

AbstractIn this paper, we present a discrete formulation of nonlinear shear- and torsion-free rods introduced by Gebhardt and Romero (Acta Mechanica 232(10):3825–3847, 2021) that uses isogeometric discretization and robust time integration. Omitting the director as an independent variable field, we reduce the number of degrees of freedom and obtain discrete solutions in multiple copies of the Euclidean space $$\left( \mathbb {R}^3\right) $$ R 3 , which is larger than the corresponding multiple copies of the manifold $$\left( \mathbb {R}^3 \varvec{\times } S^2\right) $$ R 3 × S 2 obtained with standard Hermite finite elements. For implicit time integration, we choose the same integration scheme as Gebhardt and Romero in (2021) that is a hybrid form of the midpoint and the trapezoidal rules. In addition, we apply a recently introduced approach for outlier removal by Hiemstra et al. (Comput Methods Appl Mech Eng 387:114115, 2021) that reduces high-frequency content in the response without affecting the accuracy, ensuring robustness of our nonlinear discrete formulation. We illustrate the efficiency of our nonlinear discrete formulation for static and transient rods under different loading conditions, demonstrating good accuracy in space, time and the frequency domain. Our numerical example coincides with a relevant application case, the simulation of mooring lines.

Rocking block simulation based on numerical dissipation.

In this paper, a computational approach based on numerical dissipation is proposed to simulate rocking blocks. A rocking block is idealized as a solid body interacting with its foundation through a contact-based formulation. An implicit time integration scheme with numerical dissipation, set to optimally treat dissipation in contact problems, is employed. The numerical dissipation is ruled by the time step and the rocking dissipative phenomenon at impacts is accurately predicted without any damping model. A broad numerical campaign is conducted to define a regression law in analytic form for the setting of the time step, depending on the block size and aspect ratio, the contact stiffness, as well as the coefficient of restitution selected. The so-obtained regression law appears accurate and an a posteriori validation with cases not in the training dataset confirms the effectiveness of the approach. Finally, the comparison with available experimental tests highlights the approach efficacy for free rocking and harmonic loading cases (in a deterministic sense), and for earthquake-like loading cases (in a statistical sense). It is found that rocking blocks with sizes of interest for structural engineering (e.g., cultural heritage structures) can be simulated with time steps within 10-3 ÷ 10-1s, so allowing very fast computations.

The α-generalized implicit method associated with Potra–Pták iteration for solving non-linear dynamic problems

Generally, direct time integration procedures are used for solving the equations of motion in transient analysis of structures with large displacements. In this context, we propose an algorithm that combines the α-Generalized implicit integration method with the Potra–Pták two-step iterative scheme. The free Scilab program develops a computer code for the non-linear dynamic analysis of plane frames with large displacements and rotations. The FEM corotational formulation discretizes the structures considering the Euler-Bernoulli beam theory. The null-length connection element described by the axial, translational and rotational stiffnesses simulate the behavior of the beam-column connection. Jacobi’s method and the Scilab’s spec function determine the natural frequencies. The developed program is used for modal and transient dynamic analyses of frame problems available in the literature. The numerical results show that the Potra-Pták scheme obtains approximate solutions with fewer cumulative iterations until convergence and shorter processing time compared to the standard Newton-Raphson scheme. Further-more, the results show that the type of beam-column connection affects the vibratory behavior of the structure as well as the values of its natural frequencies.

Approximation‐based implicit integration algorithm for the Simo‐Miehe model of finite‐strain inelasticity

AbstractWe propose a simple, efficient, and reliable procedure for implicit time stepping, regarding a special case of the viscoplasticity model proposed by Simo and Miehe (1992). The kinematics of this popular model is based on the multiplicative decomposition of the deformation gradient tensor, allowing for a combination of Newtonian viscosity and arbitrary isotropic hyperelasticity. The algorithm is based on approximation of precomputed solutions. Both Lagrangian and Eulerian versions of the algorithm with equivalent properties are available. The proposed numerical scheme is non‐iterative, unconditionally stable, and first order accurate. Moreover, the integration algorithm strictly preserves the inelastic incompressibility constraint, symmetry, positive definiteness, and w‐invariance. The accuracy of stress calculations is verified in a series of numerical tests, including non‐proportional loading and large strain increments. In terms of stress calculation accuracy, the proposed algorithm is equivalent to the implicit Euler method with strict inelastic incompressibility. The algorithm is implemented into MSC.MARC and a demonstration initial‐boundary value problem is solved.

High-order accurate multi-sub-step implicit integration algorithms with dissipation control for hyperbolic problems

High-order accurate multi-sub-step implicit integration algorithms with dissipation control for hyperbolic problems

Anti-symmetric and positivity preserving formulation of a spectral method for Vlasov-Poisson equations

We analyze the anti-symmetric properties of a spectral discretization for the one-dimensional Vlasov-Poisson equations. The discretization is based on a spectral expansion in velocity with the symmetrically weighted Hermite basis functions, central finite differencing in space, and an implicit Runge Kutta integrator in time. The proposed discretization preserves the anti-symmetric structure of the advection operator in the Vlasov equation, resulting in a stable numerical method. We apply such discretization to two formulations: the canonical Vlasov-Poisson equations and their continuously transformed square-root representation. The latter preserves the positivity of the particle distribution function. We derive analytically the conservation properties of both formulations, including particle number, momentum, and energy, which are verified numerically on the following benchmark problems: manufactured solution, linear and nonlinear Landau damping, two-stream instability, bump-on-tail instability, and ion-acoustic wave.

Benchmarking of implicit numerical integration methods for stiff unified constitutive equations in metal forming applications

Benchmarking of implicit numerical integration methods for stiff unified constitutive equations in metal forming applications

Linearly implicit time integration scheme of Lagrangian systems via quadratization of a nonlinear kinetic energy. Application to a rotating flexible piano hammer shank.

This paper presents a general and practical approach for nonlinear energy quadratization based on the Euler-Lagrange formulation of the physical equations. A Scalar Auxiliary Variable -like method based on a phase formulation of the equations is applied. The proposed scheme is linearly implicit, reproduces a discrete equivalent of the power balance. It is applied to a rotating and flexible piano hammer shank. An efficient solving strategy leads to a quasi explicit algorithm which shows quadratic space/time convergence.