Semi-discrete Model Research Articles

A discrete crystal model in three dimensions: The line-tension limit for dislocations

Abstract We propose a discrete lattice model of the energy of dislocations in three-dimensional crystals which properly accounts for lattice symmetry and geometry, arbitrary harmonic interatomic interactions, elastic deformations and discrete crystallographic slip on the full complement of slip systems of the crystal class. Under the assumption of diluteness, we show that the discrete energy converges, in the sense of Γ-convergence, to a line-tension energy defined on Volterra line dislocations, regarded as integral vector-valued currents supported on rectifiable curves. Remarkably, the line-tension limit is of the same form as that derived from semidiscrete models of linear elastic dislocations based on a core cutoff regularization. In particular, the line-tension energy follows from a cell relaxation and differs from the classical ansatz, which is quadratic in the Burgers vector.

Generalized Multiscale Finite Element Method for discrete network (graph) models

In this paper, we consider a time-dependent discrete network model with highly varying connectivity. The approximation by time is performed using an implicit scheme. We propose the coarse scale approximation construction of network models based on the Generalized Multiscale Finite Element Method. An accurate coarse-scale approximation is generated by solving local spectral problems in sub-networks. Convergence analysis of the proposed method is presented for semi-discrete and discrete network models. We establish the stability of the multiscale discrete network. Numerical results are presented for structured and random heterogeneous networks.

Peridynamics thermomechanical coupling simulation of damage in engineered cementitious composite-concrete bonding specimens under high temperature

The current application of Peridynamics (PD) models in addressing thermally induced failures within Engineered Cementitious Composites (ECC) and concrete bonding specimens is relatively limited. This study introduces a novel semi-discrete PD model for ECC and a random heterogeneous PD model for concrete. These models are grounded in the microscopic constitutive relationships between ECC and concrete materials and incorporate the strain softening behavior of cement-based materials. The effectiveness of the interface model is evaluated through splitting tensile tests on ECC-concrete bonding specimens, employing a series model alongside PD surface effects for material interface correction. Considering the performance degradation of materials at elevated temperatures, a PD thermo-mechanical coupling model, which integrates PD thermal diffusion and motion equations, is proposed to analyze the failure of ECC and concrete under high thermal loads. The validity of both the thermo-mechanical coupling and concrete heterogeneity models is confirmed via thermal cracking tests on heated concrete columns subjected to drilling. Lastly, considering the dynamic response of fire, the Navier-Stokes equation are reformulated into a non-local form utilizing the Peridynamic differential operator (PDDO) to simulate fire environment. A thermo-fluid-mechanical coupling model is developed, and applied to analyze ECC and concrete bonding specimens’ failure behaviors under fire scenarios.

Γ-convergence analysis of the nonlinear self-energy induced by edge dislocations in semi-discrete and discrete models in two dimensions

Abstract We propose nonlinear semi-discrete and discrete models for the elastic energy induced by a finite system of edge dislocations in two dimensions. Within the dilute regime, we analyze the asymptotic behavior of the nonlinear elastic energy, as the core-radius (in the semi-discrete model) and the lattice spacing (in the purely discrete one) vanish. Our analysis passes through a linearization procedure within the rigorous framework of Γ-convergence.

First-principles investigations of the interaction between alloying atom and dislocation and its implication to the rafting of Ni-based superalloys

The rafting of the microstructure of Ni-based superalloys significantly deteriorates the mechanical properties. According to the plastic rafting model based on the dislocation mechanism, the directional atomic diffusion induced by the interaction between the alloying atom and the interfacial dislocation is critical to the evolution of the rafting microstructure. In the present work, the interactions between the alloying atom and dislocation core in both matrix γ phase and the precipitate γ′ phase are investigated by using a first principles method in combination with the semi-discrete variational Peierls-Nabarro dislocation model. We show that the alloying atoms Al, Cr, Co, Mo, Ru, Ta, W, Re lower the (edge for Cr and Co) dislocation core energy in the γ phase, indicating that the alloying atoms are attractive to the dislocation core. For the γ′ phase, Cr, Mo, Ta, W, Re raise whereas Co and Ru lower the dislocation core energy, i.e., Cr, Mo, Ta, W, Re are repulsive but Co and Ru are attractive to the dislocation core. With the calculated interaction between the alloying atoms and the dislocation core, the influence of these alloying elements on the rafting of the superalloy is discussed.

Discrete and Semi-Discrete Multidimensional Solitons and Vortices: Established Results and Novel Findings.

This article presents a concise survey of basic discrete and semi-discrete nonlinear models, which produce two- and three-dimensional (2D and 3D) solitons, and a summary of the main theoretical and experimental results obtained for such solitons. The models are based on the discrete nonlinear Schrödinger (DNLS) equations and their generalizations, such as a system of discrete Gross-Pitaevskii (GP) equations with the Lee-Huang-Yang corrections, the 2D Salerno model (SM), DNLS equations with long-range dipole-dipole and quadrupole-quadrupole interactions, a system of coupled discrete equations for the second-harmonic generation with the quadratic (χ(2)) nonlinearity, a 2D DNLS equation with a superlattice modulation opening mini-gaps, a discretized NLS equation with rotation, a DNLS coupler and its PT-symmetric version, a system of DNLS equations for the spin-orbit-coupled (SOC) binary Bose-Einstein condensate, and others. The article presents a review of the basic species of multidimensional discrete modes, including fundamental (zero-vorticity) and vortex solitons, their bound states, gap solitons populating mini-gaps, symmetric and asymmetric solitons in the conservative and PT-symmetric couplers, cuspons in the 2D SM, discrete SOC solitons of the semi-vortex and mixed-mode types, 3D discrete skyrmions, and some others.

On semidiscrete models dominated by the heat, wave and Laplace equations

On semidiscrete models dominated by the heat, wave and Laplace equations

Efficient modelling of progressive damage due to quasi-static indentation on multidirectional laminates by a mesh orientation independent kinematically enriched continuum damage model

This work proposes a computationally efficient high-fidelity modelling framework for analysis of damage in composites subjected to quasi-static indentation. A ply-by-ply modelling approach is adopted, and a new semi-discrete continuum damage model is used for intra-ply cracking that allows for cracks to grow independent of the ply mesh pattern. This feature greatly simplifies the meshing effort since the requirement of a ply-oriented mesh and imposition of tie constraints at mesh-mismatched ply interfaces is eliminated. The model is further enhanced to realise kinematic interactions between the cracked and uncracked material domains at the constitutive level. Inter-ply delamination is simulated using cohesive elements. Through a challenging problem of static indentation on a quasi-isotropic laminate, it is shown that the model can capture solution-dependent multiple discrete ply cracks and detailed crack-delamination interaction, but with reduced computational time and complexity, compared to a reference model with ply-oriented mesh and pre-seeded cracks at known locations.

Analyzing the bending deformation of van der Waals-layered materials by a semi-discrete layer model

Van der Waals (vdW)-layered materials, such as graphite, exhibit unique mechanical properties owing to their structural and mechanical anisotropies. This study reports the development of a mechanical model that reproduces the characteristics of the nonlinear and reversible bending deformation of vdW-layered materials, while taking into account the microscopic mechanism of the discrete interlayer slips. The vdW-layered material was modeled as a stack of interacting discrete deformable layers (semi-discrete layer model), and the interlayer interaction was modeled using a cohesive zone model that reproduced the localized interlayer slip. Using the finite-element method, out-of-plane bending deformation analyses were performed on the cantilevers of the highly oriented pyrolytic graphite (HOPG) and MoTe2, and the validity of the model was verified by comparing it with the experimental results. The model accurately reproduced the loading and unloading behaviors in the experiments for the submicron HOPG cantilevers or the large nonlinear and reversible deformation with a hysteresis loop. Furthermore, the model reproduced well the characteristics of the bending experiments for the micro-MoTe2 cantilevers, or the intermittent decrease in stiffness during the loading process and deformation restoration during the unloading process. These results demonstrated that the designed semi-discrete layer model can be universally applied to reproduce the bending deformation characteristics of a variety of vdW-layered materials and can be employed to effectively elucidate the underlying deformation mechanisms.

Size Effect of Synthetic Fibre Reinforced Concrete – Investigation using a Semi-Discrete Analytical Beam Model

Abstract The size effect is a well-known characteristic of concrete structures. However, in the case of fibre-reinforced concrete (FRC), this issue is not thoroughly explored. Most design recommendations of FRC neglect the size effect or handle the behaviour of FRC structures in case of different structural sizes similar to plain concrete structures (assuming FRC is a homogeneous material). The aim of this paper is to show that the size effect of FRC can be divided, the share of the concrete matrix and the fibres in the size-dependent properties is separable. For the size effect research fifteen synthetic macro fibre reinforced concrete and six plain concrete beam specimens were prepared and tested in three different sizes and then evaluated with the semi-discrete analytical (SDA) model. The analysis of the experimental specimens has shown that the size effect significantly influences the concrete material in the case of FRC with softening material behaviour, but the residual loadbearing capacity which mainly arise from the local bridging effect of fibres is essentially independent of the structural size. It is also shown in this paper that the two defining parameters of the SDA model is independent of the structural size, so the model provides an excellent tool in case of the design of real-sized FRC structures.

A note on an algorithm studying the uniform controllability of a class of semidiscrete hyperbolic problems

We propose an algorithm based on the the technique introduced in [23]. The aim of the algorithm is to study, in a simple way, the approximation of the controls for a class of hyperbolic problems. It is well-known that, the finite-difference semi-discrete scheme for the approximation of controls can leads to high frequency numerical spurious oscillations which gives a loss of the uniform (with respect to the mesh-size) controllability property of the semidiscrete model. It is also known that an appropriate filtration of the high eigenfrequencies of the discrete initial data enable us to restore the uniform controllability property of the whole solution. But, the methods used to prove such results are very constructive and contains difficult and fine computations. As an example, which proves the effectiveness of our algorithm, we consider the case of the semidiscrete one dimensional wave equation. In this particular case, we are able to prove the uniform controllability, where the initial data are filtered in a range which contains as many modes as possibles, taking into account previous results obtained in literature (see [18]).

Imaging based on Compton scattering: model uncertainty and data-driven reconstruction methods

The recent development of scintillation crystals combined with γ-rays sources opens the way to an imaging concept based on Compton scattering, namely Compton scattering tomography. The associated inverse problem rises many challenges: non-linearity, multiple order-scattering and high level of noise. Already studied in the literature, these challenges lead unavoidably to uncertainty of the forward model. This work proposes to study exact and approximated forward models and develops two data-driven reconstruction algorithms able to tackle the inexactness of the forward model. The first one is based on the projective method called regularized sequential subspace optimization (RESESOP). We consider here a finite dimensional restriction of the semi-discrete forward model and show its well-posedness and regularization properties. The second one considers the unsupervised learning method, deep image prior, inspired by the construction of the model uncertainty in RESESOP. The methods are validated on Monte-Carlo data.

Numerical study on the effects of ply stacking and interlayer toughening on the failure modes and response of OHT laminates

The failure modes and deformation response of composite laminates are influenced by multiple factors such as stacking sequence, lamina and interlayer properties, and geometry. One of the critical failure modes in laminated composites is delamination which may lead to premature load drops. This paper investigates the effects of ply stacking and interlayer toughening through a comprehensive numerical study. The numerical framework is the latest iteration of the enhanced semi-discrete damage model (eSD2M) advanced by the authors. The present numerical framework captured the delamination type failure and pull-out failure in open-hole tensile (OHT) laminates, considering both sublaminate-level and ply-level scaled laminates. Experimental and numerical results confirmed that blocked plies can lead to premature specimen failure due to delamination. However, interlayer toughening can delay delamination and improve the structural integrity, even for ply-scaled laminates. On the other hand, interlayer toughening may have adverse effects on sublaminate-level scaled specimens. Ply-level scaled laminates may even outperform them when interlayers are slightly toughened.

Insights into plastic deformation mechanisms of austenitic steels by coupling generalized stacking fault energy and semi-discrete variational Peierls-Nabarro model

The generalized stacking fault energy (GSFE) is a key parameter to determine the plastic deformation mechanisms of austenitic steels. However, the underlying physics why the GSFE can affect the plastic deformation behaviors remains unclear. In this paper, the plastic deformation mechanisms of austenitic steels with different carbon (C) additions were investigated by coupling the GSFE with the semi-discrete variational Peierls-Nabarro (P–N) model. The internal mechanisms behind the P–N stress and plastic deformation were explained at atomic scale. It is found that the positions and contents of C atoms affect the GSFE of austenite, and thus regulate plastic deformation behaviors of austenitic steels by influencing dislocation core structure. As exemplified that with 4 ​at.%C in austenite, the intrinsic stacking fault energy increases from −433 to −264 mJ/m2, and the stacking fault width increases to 6.62b from 4.72b of FCC-Fe with b being the Burgers vector. This corresponds to the plastic deformation mechanism dominated by the ε martensitic transformation with the lattice changing from FCC to HCP. With increasing C contents to 8 ​at.%, the intrinsic stacking fault energy of austenite increases to −9.01 mJ/m2, while the stacking fault width decreases to 6.03b. The plastic deformation tends to proceed via the mechanical twinning mode. The present investigation establishes a solid foundation for clarifying the plastic deformation mechanisms of austenitic steels from the perspective of the dislocation core structure.

Mechanics of textiles used as composite preforms: A review

Textiles have been used as preforms for high-performance composite materials. Fiber architecture of the preform has a decisive effect on mechanical properties of composite. As a kind of flexible reinforcement, textile preforms can deform easily in the manufacture process of complex shaped composite structures. Mechanical behavior of textile materials is specific due to their fibrous nature. The objective of this paper is to give an overview of the literatures dedicated to the mechanics of textiles used as composite preforms. Experimental test methods and mechanical models are reviewed. The experimental tests are categorized with respect to the type of load, including tension, compaction, bending, shearing and forming. Mechanical models of textile materials are classified into macro-scale continuum, discrete, semi-discrete and hybrid element models according to the scale of investigation and the numerical simulation method. The advantages and limitations of different approaches are discussed.

Indistinguishability analysis and observer design for size-structured cell populations

The observability property of population balance equations with biomass measurements is addressed within the framework of indistinguishable trajectories. The population balance equation is described by a partial integro-differential equation which is coupled with an ordinary differential equation for the nutrient dynamics. For the semi-discretized model equations, the dynamics of indistinguishable trajectories are derived and evaluated for the special case of equal partitioning at cell division revealing that the observability property is guaranteed as long as the nutrient concentration is not depleted. Based on these results, an observer is designed and tested in simulation to estimate the cell population distribution using biomass measurements in a batch bioreactor for the case of a bi-structured population.

Spatially Adaptive Projective Integration Schemes For Stiff Hyperbolic Balance Laws With Spectral Gaps

Stiff hyperbolic balance laws exhibit large spectral gaps, especially if the relaxation term significantly varies in space. Using examples from rarefied gases and the general form of the underlying balance law model, we perform a detailed spectral analysis of the semi-discrete model that reveals the spectral gaps. Based on that, we show the inefficiency of standard time integration schemes expressed by a severe restriction of the CFL number. We then develop the first spatially adaptive projective integration schemes to overcome the prohibitive time step constraints of standard time integration schemes. The new schemes use different time integration methods in different parts of the computational domain, determined by the spatially varying value of the relaxation time. We use our analytical results to derive accurate stability bounds for the involved parameters and show that the severe time step constraint can be overcome. The new adaptive schemes show good accuracy in a numerical test case and can obtain a large speedup with respect to standard schemes.

Study on lattice discreteness effect on superdislocation core properties of Ni3Al by improved semi-discrete variational Peierls-Nabarro model

Nickel-based superalloys (NBSAs) serve as hot-end component materials of aero engines. NBSAs generally consist of γ and γ′ phases, the γ′ phase (i.e. Ni3Al) presents anomalous yield strength and non-Schmid effect, influencing the mechanical behaviors of NBSAs significantly. These phenomena are closely related to the ⟨110⟩{111} superdislocations. In the present work, based on the quasiharmonic density functional theory (DFT), the improved semi-discrete variational Peierls-Nabarro (SVPN) model is employed to systematically investigate the properties of such superdislocations, in terms of dislocation density distribution, splitting distance (i.e., anti-phase boundary APB, super-lattice intrinsic stacking fault SISF and complex stacking fault width CSF), total energy, Peierls stress and so on. It is found that the predicted superdislocation characters, especially the Peierls stress, are in good agreement with previous experimental results, considering the lattice discreteness effect. This implies that the lattice discreteness plays an important role in the total dislocation energy and Peierls stress, which has however been ignored in previous studies. Although such superdislocation may have several possible core configurations, the dissociation of four 1/6⟨112⟩ Shockley partials separated by two CSFs and an APB may be more energetically favorable at both 300 K and 1200 K. In addition, the superdislocation properties with such the core structure may be easily affected by applied stress. The present work provides an improved SVPN model with consideration of lattice discreteness to describe the superdislocation properties in Ni3Al, which may be helpful for understanding the plastic deformation behavior of the γ′ phase and NBSAs.

Agents and Events: the Objective of Action, Finite Event Duration, and Fuzzy Logic

In a traditional discrete event and queuing simulation the user defines the system resources and the possible events that the model entities (clients, moving items) will execute. In the proposed approach the entities are replaced with agents which actions are not pre-defined and rather result from the agent objectives. The resulting events are also equipped with sub-objectives to reach. The goal of this paper is to propose an event simulation paradigm, different from the traditional crisp discrete event simulation. This is a methodological and conceptual approach, rather than a practical implementation. The proposed approach is to equip the events interactions with finit duration and fuzzy logic. It is pointed out that the problem of handling of simultaneous events does not exist in such models. It is also shown that the classic crisp discrete event model is not necessarily the limit case of any sequence of semi-discrete models with event duration interval approaching zero.

Peridynamic Simulation of Dynamic Fracture Process of Engineered Cementitious Composites (ECC) with Different Curing Ages.

The mechanical properties of engineered cementitious composites (ECC) are time-dependent due to the cement hydration process. The mechanical behavior of ECC is not only related to the matrix material properties, but also to the fiber/matrix interface properties. In this study, the modeling of fiber and fiber/matrix interactions is accomplished by using a semi-discrete model in the framework of peridynamics (PD), and the time-varying laws of cement matrix and fiber/matrix interface bonding properties with curing age are also considered. The strain-softening behavior of the cement matrix is represented by introducing a correction factor to modify the pairwise force function in PD theory. The fracture damage of ECC plate from 3 to 28 days was numerically simulated by using the improved PD model to visualize the process of damage fracture under dynamic loading. The shorter the hydration time, the lower the corresponding elastic modulus, and the smaller the number of cracks generated. The dynamic fracture process of early-age ECC is analyzed to understand the crack development pattern, which provides reference for guiding structural design and engineering practice.