Implicit Gradient Research Articles

Constitutive description of snow at finite strains by the modified cam‐clay model and an implicit gradient damage formulation

AbstractSnow, characterized as a unique granular and low‐density material, exhibits intricate behavior influenced by the proximity to its melting point and its three‐phase composition. This composition entails a structured ice skeleton surrounded by voids filled with air and spread with liquid water. Mechanically, snow experiences dynamic transformations, including bonding/degradation between its grains, significant inelastic deformations, and a distinct rate sensitivity. Given snow's varied structures and mechanical strengths in natural settings, a comprehensive constitutive model is necessary. Our study introduces a pioneering formulation grounded on the modified Cam‐Clay model, extended to finite strains. This formulation is further enriched by an implicit gradient damage modeling, creating a synergistic blend that offers a detailed representation of snow behavior. The versatility of the framework is emphasized through the careful calibration of damage parameters. Such calibration allows the model to adeptly capture the effects of diverse strain rates, particularly at high magnitudes, highlighting its adaptability in replicating snow's unique mechanical responses across various conditions. Upon calibration against established experimental benchmarks, the model demonstrates a suitable alignment with observed behavior, underscoring its potential as a comprehensive tool for understanding and modeling snow behavior with precision and depth.

Optimization of Welded Joints under Fatigue Loadings

In this paper, the notch effect in weldments has been investigated, and the optimal configuration of different types of welded joints has been analysed using the implicit gradient approach. By considering this implicit gradient method, it is possible to calculate the effective stress related to fatigue damage, with the effective stress being a continuous scalar function of the real stress tensor components, even in the presence of sharp edges. Hence, the search for the optimal configuration that maximises fatigue life can be tackled as the condition of minimum effective stress obtained by changing the weld shape and geometrical parameters. Both load-carrying cruciform joints and spot welds made of steel have been considered. The structural details have been studied by modelling actual shapes without any geometric simplification. Moreover, the same numerical procedure has been considered independently of the size, shape or load condition without imposing restrictive rules on the FE mesh.

A novel FFT framework with coupled non-local elastic-plastic damage model for the thermomechanical failure analysis of UD-CF/PEEK composites

A novel FFT-based computational framework for coupling non-local elastic-plastic damage model is proposed in this paper, which is employed to accurately simulate the transverse tensile behaviors of unidirectional (UD) CF/PEEK composite materials under different temperatures. To address the distinct material properties exhibited by different microscopic constituents within the composite, two approaches are employed: (i) an integral-type regularized traction-separation damage model is applied to the interphase debonding, and (ii) an implicit gradient regularization technique for Lemaitre-type damage (matrix cracking) of the PEEK. By coupling the non-local damage model capable of accurately characterizing different failure modes within composites with an FFT computational framework, accurate predictions of the transverse tensile performance of UD-CF/PEEK composites at different temperature conditions can be achieved. Furthermore, the influence of non-local feature parameters and interphase mechanical properties on the composite's mechanical behavior is thoroughly discussed. Comparisons with experimental results affirm the accurate prediction of the transverse tensile behavior of UD-CF/PEEK composites through the proposed computational framework. Ultimately, based on this framework, the transverse tensile modulus and strength of UD-CF/PEEK composites, considering different fiber volume fractions and temperatures, are predicted, successfully demonstrating the effectiveness and applicability of the proposed approach in forecasting the mechanical behavior of composites.

Simulation of hydraulic fracture propagation based on implicit gradient non-local model

Fracture extension simulation is an important tool for studying hydraulic fracture characteristics. To solve the problem of mesh sensitivity of strain and fluid pressure during hydraulic crack propagation, the equivalent strain in the damage evolution equation was non-localized, and implicit gradient was performed. A hydraulic-mechanical coupling model based on implicit gradient non-local damage was established. Numerical simulation program was further developed. Comparison of simulation results with experimental data on ‘L’-brittle material verified the validity of the program. The fracture propagation process of hydraulic fracturing in a perforated well was simulated. On this basis, the influence of internal length scale parameters and mesh size of the non-local model on the calculation results was analyzed. The influence of mesh size on hydraulic fracture morphology and damage variables is reduced by introducing an implicit gradient nonlocal model, which effectively alleviates the localization effect in hydraulic-mechanical coupling damage simulation. This new model achieves the nonlocalization of the fluid field by introducing the relationship among porosity-permeability and deformation and damage, which avoids the mesh sensitivity of equivalent strain, damage, fluid pressure, and improves the numerical stability. It provides a new method for simulating hydraulic fracture propagation and studying the failure behavior of porous media materials.

Differential geometry-based thermodynamics derivation of isotropic and anisotropic eikonal non-local gradient (ENLG) damage models using a micromorphic media framework

The Eikonal non-local damage (ENL) approach models damage as a space-deforming phenomenon that affects the interaction distance between material points. As damage increases, interactions between points decrease, ultimately resulting in no interaction. In the integral version of such an approach, non-local interaction distances between material points are computed by solving a stationary Hamilton–Jacobi equation with a damage-dependent Riemannian metric. In the implicit gradient version of ENL models, the Riemannian metric figures in the Helmholtz equation to be solved for computing the non-local field controlling damage evolution. However, one of the main criticisms of such formulation is the lack of thermodynamics basis in its derivation. This paper presents a thermodynamics derivation of the Eikonal implicit gradient formulation based on differential geometry concepts to overcome this issue. A free-energy potential is defined considering the non-local strain as a morphological descriptor belonging to the abstract differentiable manifold (where the Riemannian metric is defined). Following a micromorphic media framework, the balance equations of the model are obtained. It is shown that the stress tensor and thermodynamic force associated with damage are the sum of standard and additional non-local contributions. It is also shown that the resulting energy dissipation is always positive, thus verifying the Clausius–Duhem inequality. After presenting all development considering second-order anisotropic continuum damage, the isotropic formulation is obtained as a particular case. Finally, a two-dimensional numerical implementation of an isotropic implicit Eikonal non-local gradient damage model is illustrated. Test cases are simulated to show the relocalization features of the considered formulation and its natural capability of naturally representing damage-to-fracture transition for high damage levels.

Micro-support effect consideration in fatigue analysis of corroded steel based on real surface geometry

The service life of steel structures is significantly reduced by corrosion. In particular, pitting corrosion leads to high local stress concentrations and can reduce the expected fatigue strength. Previous studies on corroded specimens have shown that crack development in fatigue tests is correlated with stress concentrations. To consider stress concentrations in local fatigue concepts, the so-called micro-support effects must be quantified. This can be usually considered with the theory of critical distances, including the stress averaging approach by Neuber. For stress concentrations from pitting corrosion, the applicability of this method is limited because of the large number of notches in corroded surfaces. In this study, as an alternative approach, the implicit gradient model is presented, in which the influence of the micro-support effect is considered numerically on 3D FEM models by averaging the stresses over a defined volume around the notch. The consideration of the micro-support effect revealed for most of the specimens a coincidence of the crack location with the location of the maximum notch stress. For some of the specimens, where no coincidence could be observed before micro-support consideration, the maximum stress concentration shifted to the crack location after consideration of micro-support effect. Based on this, a notch stress SN-curve for pitted steel was derived.

Meta Auxiliary Learning for Top-K Recommendation

Recommender systems are playing a significant role in modern society to alleviate the information/choice overload problem, since Internet users may feel hard to identify the most favorite items or products from millions of candidates. Thanks to the recent successes in computer vision, auxiliary learning has become a powerful means to improve the performance of a target (primary) task. Even though helpful, the auxiliary learning scheme is still less explored in recommendation models. To integrate the auxiliary learning scheme, we propose a novel meta auxiliary learning framework to facilitate the recommendation model training, i.e., user and item latent representations. Specifically, we construct two self-supervised learning tasks, regarding both users and items, as auxiliary tasks to enhance the representation effectiveness of users and items. Then the auxiliary and primary tasks are further modeled as a meta learning paradigm to adaptively control the contribution of auxiliary tasks for improving the primary recommendation task. This is achieved by an implicit gradient method guaranteeing less time complexity compared with conventional meta learning methods. Via a comparison using four real-world datasets with a number of state-of-the-art methods, we show that the proposed model outperforms the best existing models on the Top-K recommendation by 3% to 23%.

Stochastic Gradient Perturbation: An Implicit Regularizer for Person Re-Identification

Generalization of the person re-identification (ReID) model plays an important role in practical application, and we discuss a simple yet effective regularizer to improve it inspired by Adversarial Training (AT). AT has been indicated as an advanced regularizer due to its adversarial mechanism, ability to mine hard samples, and nature of data augmentation. However, serving as an augmentation-based regularizer, AT shows low diversity of the perturbation, excessive computational cost, and the optimization dilemma between adversarial robustness and accuracy for ReID task, and is thus suboptimal. To tackle these limitations and get a more effective regularizer for ReID, we rethink the nature of AT and unveil that the adversarial data augmentation is essentially reflected by gradients. Based on this, a novel implicit regularizer, named Stochastic Gradient Perturbation (SGP), is proposed, which naturally brings three merits: 1) Better diversity of the perturbation due to the proposed non-directional stochastic perturbations rather than directional adversarial perturbations. 2) Lower computational cost due to the proposed implicit gradient augmentation rather than explicitly additional data. 3) The optimization dilemma of the adversarial robustness and generalization is naturally overcome since SGP contains the adversarial gradient perturbation. Further, we put forward a perspective that the generalization and adversarial robustness may have an inter unity. Experiments on the baseline and SOTA models demonstrate powerful performances of the plugged-played SGP, and both generalization and adversarial robustness can be guaranteed.

Introduction of a recurrent neural network constitutive description within an implicit gradient enhanced damage framework

The contribution at hand presents a method for the application of Recurrent neural network based constitutive models within a coupled field Finite Element Analysis. Thereby, an additional scalar field is coupled to the displacement field and evolved by a scalar variable at the integration points. By employing implicit gradient enhancement as regularization framework, the representation of softening material behavior is possible without the generation of localization phenomena. The feasibility of this approach is shown for an anisotropic constitutive description based on the microplane model, by employing a self-adversarial training scheme for enhanced robustness which was proposed to work for arbitrary inelastic materials. Its capabilities and limitations during application in Finite Element Analysis are evaluated and discussed for two numerical simulations.

Anisotropic damage behavior in fiber-based materials: Modeling and experimental validation

This study presents a thermodynamically consistent continuum damage model for fiber-based materials that combines elastoplasticity and damage mechanisms to simulate the nonlinear mechanical behavior under in-plane loading. The anisotropic plastic response is characterized by a non-quadratic yield surface composed of six sub-surfaces, providing flexibility in defining plastic properties and accuracy in reproducing material response. The damage response is modeled based on detailed uniaxial monotonic and cyclic tension-loaded experiments conducted on specimens extracted from a paper sheet in various directions. To account for anisotropic damage, we propose a criterion consisting of three sub-surfaces representing tension damage in the in-plane material principal directions and shear direction, where the damage onset is determined through cyclic loading tests. The damage evolution employs a normalized fracture energy concept based on experimental observation, which accommodates an arbitrary uniaxial loading direction. To obtain a mesh-independent numerical solution, the model is regularized using the implicit gradient enhancement by utilizing the linear heat equation solver available in commercial finite-element software. The study provides insights into the damage behavior of fiber-based materials, which can exhibit a range of failure modes from brittle-like to ductile, and establishes relationships between different length measurements.

Simulation of hydrogen embrittlement of steel using mixed nonlocal finite elements

A new simulation strategy to model hydrogen embrittlement based on a multi-field finite element using displacements, pressure, volume variation, nonlocal damage variables and lattice hydrogen concentration as unknowns is proposed. The material is described using a modified GTN model, which includes the description of hydrogen enhanced decohesion (HEDE). The finite element problem is solved using a fully implicit formulation. Numerical problems related to volumetric locking are solved using a mixed pressure/volume variation formulation. The use of the mixed formulation allows a straightforward evaluation of the pressure gradient, which drives hydrogen diffusion. Mesh size dependence is solved using an implicit gradient nonlocal formulation. Two variables are used to represent damage: the plastic volume variation and the accumulated plastic strain, controlling nucleation. The model is used to simulate an existing experimental database (Moro et al., 2010; Briottet et al., 2012) including tensile, fracture toughness and pressurized disk tests. The model, after adjusting the various coefficients, can represent the main experimental findings: the effect of deformation rate on the failure of tensile specimens, the transition from surface to internal fracture with increasing deformation rate, the sharp toughness drop under hydrogen (CT specimen), the fracture location and the effect of the pressurization rate in the case of the tests on disks.

Nonlocal implicit gradient enhancements for strain localization informed by controllability criteria for plastic solids

Nonlocal implicit gradient enhancements are widely used to suppress mesh dependency in simulations involving strain localization. For example, nonlocality is often introduced through internal variables that account for possible material softening. This work, however, shows that this approach may become ineffective if the solid displays plastic non-normality (i.e., non-associated plastic flow). For this purpose, we consider an over-nonlocal formulation and mathematically inspect the conditions at which regularization is lost in the presence of plastic non-normality. Specifically, such loss of regularization is linked to the loss of uniqueness and/or existence of the incremental plastic response that is kinematically compatible with the development of a deformation band. By doing so, we find a lower limit for the admissibility of the parameters controlling the effectiveness of nonlocal implicit gradient regularization. Furthermore, we show that such a lower limit is regulated by a plastic modulus reflecting the loss of controllability of the constitutive response, and, hence, depends on the degree of plastic non-normality. We also derive a closed-form expression relating the thickness of the deformation band to both the controllability modulus and gradient regularization constants, which suggests that the thickness of the process zone may change in response to the prevailing plastic flow characteristics and evolve during active plastic deformation. The proposed nonlocal enhancement is applied to a non-associated elasto-plastic model for porous sedimentary rocks, which is capable of displaying both shear-dominated and compaction-dominated bands. Numerical simulations reveal that effective regularization can be enforced only when the over-nonlocal weighting coefficient is larger than the above-mentioned lower limit.

A non-local approach for Reissner–Mindlin shell elements in dynamic simulations: Application with a Gurson model

Numerical prediction of ductile tearing during a car crash is necessary to ensure the structural integrity of the vehicle. Recently, encouraging results were obtained by Davaze et al., (2021) with a non-local damage approach, compatible with dynamic explicit simulations and preserving parallel computation: an implicit gradient model enabled to reproduce experimental results with DP450 steel sheets with 3D solid elements. However, in the automotive industry, thin sheets of metals are mostly modeled with shell elements for computation time efficiency. Shell elements have a more complex formulation: their kinematic description is enriched accounting for displacements but also rotations, leading to two different strain contributions: membrane strain and bending strain. The transverse strain is computed independently in a post-processing step from local internal variables with a zero normal stress assumption. This formulation is known to lead to strain localization after the onset of necking, while this is not the case with 3D solid elements. The problem is even more challenging when considering shells in association with a softening material accounting for damage as this also leads to strong mesh dependency, as with 3D solid elements. This complex problem is solved using non-local formulations for the material behavior. The proposed solution accounts for both in-plane and in-thickness damage gradients. It consists of a two-step regularization procedure introducing nested wire elements in the shell thickness. Using non-local variables to evaluate the transverse strain enables to obtain a continuous spatial distribution for strains, in agreement with the 3D solid case. An updated Lagrangian formulation is used to account for large strains. The performance of the strategy is demonstrated in three test cases, with a comparison with an experimental result for one of them. The results obtained following the proposed strategy lead are equivalent to the ones obtained with the non-local method proposed by Davaze et al., (2021) and are thus consistent with the experimental data.

Scalable Bayesian Meta-Learning through Generalized Implicit Gradients

Meta-learning owns unique effectiveness and swiftness in tackling emerging tasks with limited data. Its broad applicability is revealed by viewing it as a bi-level optimization problem. The resultant algorithmic viewpoint however, faces scalability issues when the inner-level optimization relies on gradient-based iterations. Implicit differentiation has been considered to alleviate this challenge, but it is restricted to an isotropic Gaussian prior, and only favors deterministic meta-learning approaches. This work markedly mitigates the scalability bottleneck by cross-fertilizing the benefits of implicit differentiation to probabilistic Bayesian meta-learning. The novel implicit Bayesian meta-learning (iBaML) method not only broadens the scope of learnable priors, but also quantifies the associated uncertainty. Furthermore, the ultimate complexity is well controlled regardless of the inner-level optimization trajectory. Analytical error bounds are established to demonstrate the precision and efficiency of the generalized implicit gradient over the explicit one. Extensive numerical tests are also carried out to empirically validate the performance of the proposed method.

A gradient-based non-local GTN model: Explicit finite element simulation of ductile damage and fracture

Non-local models have over the years been established as an effective approach to solve the pathological mesh dependency problem observed in finite element simulation of strain-softening materials. This paper presents the formulation, implementation, and application of a gradient-based non-local Gurson–Tvergaard–Needleman (GTN) model for explicit finite element analysis. The porosity is taken as the non-local variable where the increment in porosity is averaged over the volume using an implicit gradient model. The gradient model is implemented in Abaqus/Explicit by utilising the coupled thermal–mechanical solver, which proves to be both a simple and computationally efficient approach. Due to the use of an explicit integration scheme, a transient term is introduced to the partial differential equation of the gradient formulation. The non-local GTN model is compared to the local counterpart for increasing mesh refinements using a plane strain shear band specimen, a plane strain tension specimen, and a plane strain compact tension specimen. The proposed approach can remedy the pathological mesh dependency problem. For the plane strain tension specimen, it is shown that the non-local GTN model will preserve the fracture mode (slant versus cup–cup fracture mode) when the mesh is refined. The non-local GTN model is also able to predict the same fracture mode as observed in ductile tearing experiments. However, non-local averaging can over-smooth the fields and exclude the slant fracture mode from occurring if the material length Lc is too large.

A new implicit gradient damage model based on energy limiter for brittle fracture: Theory and numerical investigation

We present a general form of the gradient-enhanced damage theory and its numerical implementation using finite element method (FEM) in modeling quasi-static brittle crack growth in one- (1D), two- (2D) and three-dimensional (3D) bodies. Coupled equations of the equilibrium and a new implicit gradient damage formulation are introduced to govern the deformation of the solid and evolution of the damage. The resulting nonlocal damage evolution equation featuring the growth of diffusive crack is integrated with a characteristic length scale to eliminate the common mesh-bias issue in FEM implementation. In contrast to the traditional gradient-enhanced damage approaches, the nonlocal damage field here is defined as the primary variable of the damage evolution equation without interpolation mismatch between the displacement and nonlocal damage fields. For derivation of the material constitutive law and local damage parameter, a novel strain energy density (SED) function based on the energy limiter theory for brittle crack growth problems under small strain regime is introduced. To further improve the performance of the developed model, an initial SED threshold, which is used for determining the critical point when damage starts to initiate in the material, is integrated into the novel energy limiter theory. For preventing nonphysical failure in compression domains, the spectral decomposition technique for the strain tensor is adopted to split the reference SED. With integrating the energy limiter into the developed theory, unlike the conventional nonlocal damage theories where the interpretation of the length scale is still ambiguous, the developed nonlocal damage model defines the length scale parameter as the problem-dependent factor. The performance and ability of the proposed model are demonstrated via a set of representative numerical examples in 1D, 2D and 3D fracture problems.

Optical neural network via loose neuron array and functional learning

This research proposes a deep-learning paradigm, termed functional learning (FL), to physically train a loose neuron array, a group of non-handcrafted, non-differentiable, and loosely connected physical neurons whose connections and gradients are beyond explicit expression. The paradigm targets training non-differentiable hardware, and therefore solves many interdisciplinary challenges at once: the precise modeling and control of high-dimensional systems, the on-site calibration of multimodal hardware imperfectness, and the end-to-end training of non-differentiable and modeless physical neurons through implicit gradient propagation. It offers a methodology to build hardware without handcrafted design, strict fabrication, and precise assembling, thus forging paths for hardware design, chip manufacturing, physical neuron training, and system control. In addition, the functional learning paradigm is numerically and physically verified with an original light field neural network (LFNN). It realizes a programmable incoherent optical neural network, a well-known challenge that delivers light-speed, high-bandwidth, and power-efficient neural network inference via processing parallel visible light signals in the free space. As a promising supplement to existing power- and bandwidth-constrained digital neural networks, light field neural network has various potential applications: brain-inspired optical computation, high-bandwidth power-efficient neural network inference, and light-speed programmable lens/displays/detectors that operate in visible light.

A nonlocal microplane approach to model textile reinforced concrete at finite deformations

The contribution at hand focuses on the introduction of a new constitutive approach at finite deformations to represent an initially anisotropic material behavior accounting for fibers within a microplane model. To capture the specific behavior of concrete structures, a formulation is chosen, which is characterized by damage and plasticity phenomena. To achieve a consistent formulation for finite strain plasticity, a description relative to the plastic intermediate configuration is adopted. Therefore, a simulation of fiber reinforced concrete is enabled at large strains, which overcomes downsides of previous formulations. To overcome numerical instabilities, which can occur in softening phenomena, an implicit gradient enhancement is utilized. The novel formulations, within the microplane framework, enable a sufficiently close approximation of fiber reinforced concrete structures. In the numerical examples, it is demonstrated, that the newly introduced fiber formulation leads to physically meaningful results. Furthermore, a validation of the proposed model is carried out for concrete undergoing more than 30 % strain. Additionally, the fiber formulation is compared to tension tests carried out on textile reinforced concrete. It can be demonstrated, that the proposed formulations can be applied for different volume percentages of fiber reinforcement in concrete structures. As a consequence, it is now possible to simulate fiber reinforced concrete structures consistently for arbitrarily large strains.

Fatigue & Fracture of Engineering Materials & Structures Virtual Special Issue: Data science and machine learning for fatigue and fracture assessment

<scp>Fatigue &amp; Fracture of Engineering Materials &amp; Structures Virtual Special Issue</scp>: Data science and machine learning for fatigue and fracture assessment

Implicit Gradients Based Conservative Numerical Scheme for Compressible Flows

This paper introduces a novel approach to compute the numerical fluxes at the cell boundaries for a cell-centered conservative numerical scheme. Explicit gradients used in deriving the reconstruction polynomials are replaced by high-order gradients computed by compact finite differences, referred to as implicit gradients in this paper. A problem-independent shock capturing approach via Boundary Variation Diminishing (BVD) algorithm is used to suppress oscillations for the simulation of flows with shocks and material interfaces. Several numerical test cases are carried out to verify the proposed method’s capability using the implicit gradient method for compressible flows.