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PurposeThe purpose of this study is to show how gradual strength degradation of metal beams under cyclic bending up to fatigue failure is simulated based on a new elastoplasticity model free of any yield criterion.Design/methodology/approachA new approach is proposed toward accurately and explicitly prescribing evolution of non-uniform stress distribution on beam cross-section under cyclic bending and, as such, gradual degradation of the bending strength can be directly determined.FindingsExplicit results for the bending response in a whole cyclic process up to fatigue failure are obtained and the fatigue characteristic curve is for the first time simulated directly between the curvature amplitude and the cycle number to failure.Originality/valueFirst, explicit and accurate determination of the non-uniform stress distribution on beam cross-section is achieved with asymptotic softening effects. Second, degradation of the bending strength can be directly deduced cycle by cycle. Finally, the relationship between the bending moment and the curvature is calculated using new and efficient numerical algorithms, thus bypassing usual time-consuming calculations with finite element procedures. Numerical results are presented and in good agreement with experimental data.

In this in vitro study, the restoration of acid-eroded enamel surface morphology and anti-wear properties under two conditions, mono-remineralization (treated with remineralization alone) and impact-remineralization (treated with cyclic impact followed by remineralization), are characterized to determine the effect of occlusal loading on enamel remineralization. Compared with the mono-remineralized surface, the impact-remineralized surface demonstrates better anti-wear performance, as manifested by a higher hardness and elastic modulus, as well as a lower friction coefficient and wear volume. Loading on the eroded enamel surface induces the fragmentation of hydroxyapatite nanoparticles, which aids crystal deposition and fusion during subsequent remineralization. In summary, owing to the enamel microstructure, occlusal loading can promote the restoration of enamel anti-wear properties by enhancing remineralization. Remineralization enhancement through occlusal-loading-induced nanoparticle fragmentation plays a significant role in preventing human teeth from excessive wear.

Toward accurately simulating both hardening and softening effects for metals up to failure, a new finite strain elastoplastic J2-flow model is proposed with the yield strength therein as a function of the plastic work in the explicit form. With no need to identify any adjustable parameters, the uniaxial stress-strain response predicted from this new model is shown to automatically and accurately match any given data from monotonic uniaxial extension tests of bars. As such, the objectives in three respects are achieved for the first time, i.e., (i) both the hardening and softening effects up to failure can be simulated in the sense of matching test data with no errors, (ii) the usual tedious implicit procedures toward identifying numerous unknown parameters need not be involved and can be totally bypassed, and (iii) the model applicability can be ensured in a broad sense for various metallic materials with markedly different transition effects from hardening to softening. With the new model, the complete response features of stretched bars and twisted tubes up to failure are studied, including the failure effects of bars under monotonic extension and tubes under monotonic torsion and, furthermore, the fatigue failure effects of bars under cyclic loading. The results show accurate agreement with the uniaxial data, and the results for both the shear stress and the normal stress at the finite torsion display realistic hardening-to-softening transition effects for the first time.

A new finite strain elatoplastic J2-flow model with coupling effects of both isotropic and anisotropic hardening is proposed with the co-rotational logarithmic rate. In terms of certain single-variable shape functions representing uniaxial loading and unloading curves, explicit multi-axial expressions for the three hardening quantities incorporated in the new model proposed are derived in unified forms for the purpose of automatically and accurately simulating complex pseudoelastic-to-plastic transition effects of shape memory alloys (SMAs) under multiple loading-unloading cycles. Numerical examples show that with only a single parameter of direct physical meaning for each cycle, accurate and explicit simulations may be achieved for extensive data from multiple cycle tests.

A new finite strain elastoplastic $$J_{\mathrm {2}}$$ -flow model is established with an explicit formulation of work-hardening and softening effects up to eventual failure, in which both a new flow rule free of yielding and an asymptotically vanishing stress limit are incorporated. The novelties of this new model are as follows: (i) Fatigue failure effects under repeated loading conditions with either constant or varying amplitudes are automatically characterized as inherent response features; (ii) neither additional damage-like variables nor failure criteria need to be involved; and (iii) both high- and low-cycle fatigue effects may be simultaneously treated. A fast and efficient algorithm of high accuracy is proposed for directly simulating high- and medium–high-cycle fatigue failure effects under repeated loading conditions. Toward this goal, a direct and explicit relationship between the fatigue life and the stress amplitude is obtained by means of explicit and direct procedures of integrating the coupled elastoplastic rate equations for any given number of loading–unloading cycles with varying stress amplitudes. Numerical examples suggest that the new algorithm is much more fast and efficient than usual tedious and very time-consuming integration procedures.

In this study, the protective effects of two food hydrocolloids, Xanthan gum and Arabic gum, on dental erosion are investigated from the perspective of the nanomechanical properties and microtribological behavior of acid-eroded enamel. Enamel specimens prepared from extracted human teeth were immersed in citric acid solution (CAS), CAS with 0.03% w/v Xanthan gum and CAS with 0.03% w/v Arabic gum, respectively, for 10 min to obtain three groups of eroded specimens. The nanomechanical properties and microtribological behavior of enamel were examined using nano-indentation/scratch techniques. The results show that compared with Arabic gum, Xanthan gum inhibits enamel surface demineralization and acid permeation more effectively because of a more uniform and denser adsorption on the surface of the enamel. The impairment of the nanomechanical and microtribological properties of the enamel surface by acid erosion is mitigated more significantly by adding trace amounts of Xanthan gum than Arabic gum. In summary, adding trace food hydrocolloids reduces enamel surface demineralization and inhibits acid permeation to mitigate the influence of erosion on the mechanical and tribological properties of enamel. The adsorption state of food hydrocolloids is the determining factor in the permeability of acid agents into the enamel and plays a significant role in preventing dental erosion.

New finite elastoplastic J2-flow equations are proposed for the purpose of explicitly simulating continuous transitions from plastic to pseudo-elastic effects of SMAs under multiple loading-unloading cycles. Novelties of such models are as follows: (i) New hardening effects are incorporated by introducing interplay between the changing of the yield surface radius and the moving of the yield surface center; (ii) extensive experimental data may be accurately simulated for any given number of loading-unloading cycles; (iii) the complicated task for simulating extensive data may be accomplished by independently fitting certain single-variable functions; and (iv) such single-variable functions may be presented in explicit forms for the purpose of automatically, accurately fitting extensive data, thus bypassing uncertainties and undue complexities usually involved in iteratively solving a coupled system of nonlinear constitutive rate equations toward identifying numerous unknown parameters. Numerical examples for model validation are presented and in good agreement with stress-strain data up to 20 loading-unloading cycles given in literature.

Finite strain elastoplastic J2-flow equations are established toward the purpose of automatically, accurately simulating pseudo-elastic effects of SMAs. The uniaxial responses derived from these equations in each loading-unloading cycle exactly produce a closed stress-strain curve of any given shape. Then, any given test data for pseudo-elastic hysteresis loops of SMAs may be accurately fitted by means of a new technique for combining linear spline functions into a unified, smooth interpolating function, in a sense with no need to identify any unknown parameters.

Most turbulence models used in Reynolds-averaged Navier-Stokes (RANS) simulations are partial differential equations (PDE) that describe the transport of turbulent quantities. Such quantities include turbulent kinetic energy for eddy viscosity models and the Reynolds stress tensor (or its anisotropy) in differential stress models. However, such models all have limitations in their robustness and accuracy. Inspired by the successes of machine learning in other scientific fields, researchers have developed data-driven turbulence models. Recently, a nonlocal vector-cloud neural network with embedded invariance was proposed, with its capability demonstrated in emulating passive tracer transport in laminar flows. Building upon this success, we use nonlocal neural network mapping to model the transport physics in the k-epsilon model and couple it to RANS solvers, leading to a PDE-free eddy-viscosity model. We demonstrate the robustness and stability of the RANS solver with a neural network-based turbulence model on flows over periodic hills of parameterized geometries. Our work serves as a proof of concept for using a vector-cloud neural network as an alternative to traditional turbulence models in coupled RANS simulations. The success of the coupling paves the way for neural network-based emulation of Reynolds stress transport models.

In this work, we propose using an ensemble Kalman method to learn a nonlinear eddy viscosity model, represented as a tensor basis neural network, from velocity data. Data-driven turbulence models have emerged as a promising alternative to traditional models for providing closure mapping from the mean velocities to Reynolds stresses. Most data-driven models in this category need full-field Reynolds stress data for training, which not only places stringent demand on the data generation but also makes the trained model ill-conditioned and lacks robustness. This difficulty can be alleviated by incorporating the Reynolds-averaged Navier–Stokes (RANS) solver in the training process. However, this would necessitate developing adjoint solvers of the RANS model, which requires extra effort in code development and maintenance. Given this difficulty, we present an ensemble Kalman method with an adaptive step size to train a neural-network-based turbulence model by using indirect observation data. To our knowledge, this is the first such attempt in turbulence modelling. The ensemble method is first verified on the flow in a square duct, where it correctly learns the underlying turbulence models from velocity data. Then the generalizability of the learned model is evaluated on a family of separated flows over periodic hills. It is demonstrated that the turbulence model learned in one flow can predict flows in similar configurations with varying slopes.

In view of the long stagnation in traditional turbulence modeling, researchers have attempted using machine learning to augment turbulence models. This paper presents some of the recent progresses in our group on augmenting turbulence models with physics-informed machine learning. We also discuss our works on ensemble-based field inversion to provide training data for constructing machine learning models. Future and on-going research efforts are introduced.

Purpose A direct and unified approach is proposed toward simultaneously simulating large strain elastic behaviors of gellan gels with different gellan polymer concentrations. The purpose of this paper is to construct an elastic potential with certain parameters of direct physical meanings, based on well-designed invariants of Hencky’s logarithmic strain. Design/methodology/approach For each given value of the concentration, the values of the parameters incorporated may be determined in the sense of achieving accurate agreement with large strain uniaxial extension and compression data. By means of a new interpolating technique, each parameter as a function of the concentration is then obtained from a given set of parameter values for certain concentration values. Findings Then, the effects of gellan polymer concentrations on large strain elastic behaviors of gellan gels are studied in demonstrating how each parameter relies on the concentration. Plane-strain (simple shear) responses are also presented for gellan gels with different polymer concentrations. Originality/value A direct, unified approach was proposed toward achieving a simultaneous simulation of large elastic strain behaviors of gellan gels for different gellan polymer concentrations. Each parameter incorporated in the proposed elastic potential will be derived as a function of the polymer concentration in an explicit form, in the very sense of simultaneously simulating large strain data for different concentrations.

Deformable micro-continua of highly localized nature are found to exactly exhibit all quantum effects commonly known for quantum entities at microscopic scale. At every instant, the spatial configuration of each such micro-continuum is prescribed by four spatial distributions of the mass, the velocity, the internal stress, and the intrinsic angular momentum. The deformability features of such micro-continua in response to all configuration changes are identified with a constitutive equation that specifies how the internal stress responds to the mass density field. It is shown that these microcontinua are endowed with the following unique response features: (i) the coupled system of the nonlinear field equations governing their dynamic responses to any given force and torque fields is exactly reducible to a linear dynamic equation governing a complex field variable; (ii) this fundamental dynamic equation and this complex field variable are just the Schrodinger equation and the complex wave function in quantum theory; and, accordingly, (iii) the latter two and all quantum effects known for quantum entities are in a natural and unified manner incorporated as the inherent response features of the micro-continua discovered, thus following objective and deterministic response patterns for quantum entities, in which the physical origins and meanings of the wave function and the Schrodinger equation become self-evident and, in particular, any probabilistic indeterminacy becomes irrelevant.

A new approach is proposed to characterize the work-hardening behavior of metals based on the stress–strain data from uniaxial extension testing. With this new approach, the yield strength as a function of the plastic work can be determined by directly fitting a well-chosen single-variable shape function to any given uniaxial data from the initial yielding up to the strength limit, in an explicit sense with no need to carry out the usual tedious trial-and-error procedures in treating nonlinear elastoplastic rate equations toward identifying numerous unknown parameters. Numerical examples show that the simulation results with the new approach are in accurate agreement with the test data.

Constitutive and closure models play important roles in computational mechanics and computational physics in general. Classical constitutive models for solid and fluid materials are typically local, algebraic equations or flow rules describing the dependence of stress on the local strain and/or strain-rate. Closure models such as those describing Reynolds stress in turbulent flows and laminar–turbulent transition can involve transport PDEs (partial differential equations). Such models play similar roles to constitutive relation, but they are often more challenging to develop and calibrate as they describe nonlocal mappings and often contain many submodels. Inspired by the structure of the exact solutions to linear transport PDEs, we propose a neural network representing a region-to-point mapping to describe such nonlocal constitutive models. The range of nonlocal dependence and the convolution structure are derived from the formal solution to transport equations. The neural network-based nonlocal constitutive model is trained with data. Numerical experiments demonstrate the predictive capability of the proposed method. Moreover, the proposed network learned the embedded submodel without using data from that level, thanks to its interpretable mathematical structure , which makes it a promising alternative to traditional nonlocal constitutive models.

Constitutive models are widely used for modeling complex systems in science and engineering, where first-principle-based, well-resolved simulations are often prohibitively expensive. For example, in fluid dynamics, constitutive models are required to describe nonlocal, unresolved physics such as turbulence and laminar–turbulent transition. However, traditional constitutive models based on partial differential equations (PDEs) often lack robustness and are too rigid to accommodate diverse calibration datasets. We propose a frame-independent, nonlocal constitutive model based on a vector-cloud neural network that can be learned with data. The model predicts the closure variable at a point based on the flow information in its neighborhood. Such nonlocal information is represented by a group of points, each having a feature vector attached to it, and thus the input is referred to as vector cloud. The cloud is mapped to the closure variable through a frame-independent neural network, invariant both to coordinate translation and rotation and to the ordering of points in the cloud. As such, the network can deal with any number of arbitrarily arranged grid points and thus is suitable for unstructured meshes in fluid simulations . The merits of the proposed network are demonstrated for scalar transport PDEs on a family of parameterized periodic hill geometries. The vector-cloud neural network is a promising tool not only as nonlocal constitutive models and but also as general surrogate models for PDEs on irregular domains.

New finite-strain elastoplastic constitutive equations are proposed toward a direct approach for simulating both monotonic and cyclic failure effects of metals. Novelties in several respects are incorporated in these new equations: (a) No yield conditions need be imposed, and therefore smooth transition from an elastic to a plastic state may be ensured without involving strong discontinuities in elastoplastic tangent moduli; (b) Complex forms of usual elastoplastic equations assuming the loading–unloading conditions may be substantially simplified in a more realistic sense of representing elastoplastic behavior of metals; (c) Failure effects in a broad case of multi-axial loading conditions may be automatically predicted as inherent constitutive features, without involving any additional variables and any ad hoc failure criteria; and (d) A new evolution equation for the back stress is introduced for effective characterization of both hardening and softening effects associated with strain-induced anisotropy. A simultaneous simulation of monotonic and cyclic failure effects of metal tubes under torsional loadings is for the first time presented for both free- and fixed-end torsion with finite rotation. Results are obtained for coupling effects of finite rotation and strain-induced anisotropy. Model predictions compare well with experimental data.

Innovative elastoplastic J 2 − flow models with nonlinear combined hardening are proposed for the purpose of simulating multi-axial thermo-coupled fatigue failure for metals and alloys, in a direct sense without involving any usual damage-like variables and any ad hoc failure criteria. As contrasted with usual models, the yield condition and the loading-unloading conditions need not be imposed as extrinsic coercive conditions but may be automatically incorporated into a free, smooth flow rule in a more realistic sense. Novel results in three respects are available directly from model predictions, namely, (i) complex features of thermo-coupled fatigue failure may be automatically represented by simple asymptotic properties of the hardening functions introduced, (ii) critical failure states may be derived with a criterion in unified form, and, in particular, (iii) direct procedures may be established for determining fatigue lives under either stress cycles or thermal cycles. Simulation results in the simplest cases of the proposed model are in good agreement with experimental data.

A hyperelastic model is proposed to characterize large elastic deformation behaviors of soft solids. Novelties in four respects are incorporated in this model, namely, (i) general compressible deformations may be treated and, accordingly, the usually-assumed incompressibility constraint and associated issues may be bypassed; (ii) applicability for all kinds of material samples may be automatically ensured, in the sense that any given test data for three deformation modes may be exactly fitted, including uniaxial, equi-biaxial and plane-strain extension; (iii) model parameters of direct physical meanings may be provided to characterize both strain-stiffening and softening effects; and (iv) breaking behavior may be simulated. Numerical results for model validation are in good agreement with Treloar’s classic data for rubbers and with extensive data for various kinds of polymer gels.

The objective of this study is two-fold. Firstly, new finite strain elastoplasticity models are proposed from a fresh standpoint to achieve a comprehensive representation of thermomechanical behavior of metals and alloys over the whole deformation range up to failure. As contrasted with the usual elastoplasticity models, such new models of much simpler structure are totally free, in the sense that both the yield condition and the loading–unloading conditions need not be introduced as extrinsic coercive conditions but are automatically incorporated as inherent constitutive features into the models. Furthermore, the new models are shown to be thermodynamically consistent, in a further sense that both the specific entropy function and the Helmholtz free energy function may be presented in explicit forms, such that the thermodynamic restriction stipulated by Clausius–Duhem inequality for the intrinsic dissipation may be identically satisfied. Secondly, it is then demonstrated that the thermo-coupled fatigue failure behavior under combined cyclic changes of stress and temperature may be derived as direct consequences from the new models. This novel result implies that the new model can directly characterize the thermo-coupled fatigue failure behavior of metals and alloys, without involving any usual damage-like variables as well as any ad hoc additional criteria for failure. In particular, numerical examples show that, under cyclic changes of temperature, the fatigue characteristic curve of fatigue life versus temperature amplitude may be obtained for the first time from model prediction both in the absence and in the presence of stress. Results are in agreement with the salient features of metal fatigue failure.

Inverse problems in computational mechanics consist of inferring physical fields that are latent in the model describing some observable fields. For instance, an inverse problem of interest is inferring the Reynolds stress field in the Navier–Stokes equations describing mean fluid velocity and pressure. The physical nature of the latent fields means they have their own set of physical constraints, including boundary conditions. The inherent ill-posedness of inverse problems, however, means that there exist many possible latent fields that do not satisfy their physical constraints while still resulting in satisfactory agreement in the observation space. These physical constraints must therefore be enforced through the problem formulation. So far there has been no general approach to enforce boundary conditions on latent fields in inverse problems in computational mechanics, with these constraints often simply ignored. In this work we demonstrate how to enforce boundary conditions in Bayesian inversion problems by choice of the statistical model for the latent fields. Specifically, this is done by modifying the covariance kernel to guarantee that all realizations satisfy known values or derivatives at the boundary. As a test case the problem of inferring the eddy viscosity in the Reynolds-averaged Navier–Stokes equations is considered. The results show that enforcing these constraints results in similar improvements in the output fields but with latent fields that behave as expected at the boundaries.

Reynolds-averaged Navier–Stokes (RANS) simulations with turbulence closure models continue to play important roles in industrial flow simulations. However, the commonly used linear eddy-viscosity models are intrinsically unable to handle flows with non-equilibrium turbulence (e.g. flows with massive separation). Reynolds stress models, on the other hand, are plagued by their lack of robustness. Recent studies in plane channel flows found that even substituting Reynolds stresses with errors below 0.5 % from direct numerical simulation databases into RANS equations leads to velocities with large errors (up to 35 %). While such an observation may have only marginal relevance to traditional Reynolds stress models, it is disturbing for the recently emerging data-driven models that treat the Reynolds stress as an explicit source term in the RANS equations, as it suggests that the RANS equations with such models can be ill-conditioned. So far, a rigorous analysis of the condition of such models is still lacking. As such, in this work we propose a metric based on local condition number function for a priori evaluation of the conditioning of the RANS equations. We further show that the ill-conditioning cannot be explained by the global matrix condition number of the discretized RANS equations. Comprehensive numerical tests are performed on turbulent channel flows at various Reynolds numbers and additionally on two complex flows, i.e. flow over periodic hills, and flow in a square duct. Results suggest that the proposed metric can adequately explain observations in previous studies, i.e. deteriorated model conditioning with increasing Reynolds number and better conditioning of the implicit treatment of the Reynolds stress compared to the explicit treatment. This metric can play critical roles in the future development of data-driven turbulence models by enforcing the conditioning as a requirement on these models.

The emerging push of the differentiable programming paradigm in scientific computing is conducive to training deep learning turbulence models using indirect observations. This paper demonstrates the viability of this approach and presents an end-to-end differentiable framework for training deep neural networks to learn eddy viscosity models from indirect observations derived from the velocity and pressure fields. The framework consists of a Reynolds-averaged Navier–Stokes (RANS) solver and a neural-network-represented turbulence model, each accompanied by its derivative computations. For computing the sensitivities of the indirect observations to the Reynolds stress field, we use the continuous adjoint equations for the RANS equations, while the gradient of the neural network is obtained via its built-in automatic differentiation capability. We demonstrate the ability of this approach to learn the true underlying turbulence closure when one exists by training models using synthetic velocity data from linear and nonlinear closures. We also train a linear eddy viscosity model using synthetic velocity measurements from direct numerical simulations of the Navier–Stokes equations for which no true underlying linear closure exists. The trained deep-neural-network turbulence model showed predictive capability on similar flows.

It is reported that there exist deformable media which display quantum effects just as quantum entities do. As such, each quantum entity usually treated as a point particle may be represented by a deformable medium, the dynamic behavior of which is prescribed by four dynamic state variables, including mass density, velocity, internal pressure, and intrinsic angular momentum. In conjunction with the finding of the characteristic equation characterizing the physical nature of such media, it is found that a complex field quantity may be introduced to uncover a perhaps unexpected correlation, i.e., the governing dynamic equations for such media may be exactly reduced to the Schrodinger equation, from which the closed-form solutions for all the four dynamic state variables can be obtained. It turns out that this complex field quantity is just the wavefunction in the Schrodinger equation. Moreover, the dynamic effects peculiar to spin are derivable as direct consequences. It appears that these results provide a missing link in quantum theory, in the sense of disclosing the physical origin and nature of both the wavefunction and the wave equation. Now, the inherent indeterminacy in quantum theory may be rendered irrelevant. The consequences are explained for certain long-standing fundamental issues.

A direct approach is proposed to obtain new multi-axial elastic potentials for incompressible soft solids. Results are presented with novelties in three respects, namely (i) any given benchmark test data for three deformation modes may be exactly fitted, including uniaxial, equi-biaxial, and plane-strain extension; (ii) model parameters of direct physical meanings may be provided to represent both the strain-stiffening effect and failure behavior; and (iii) error estimation may be established for all possible deformation modes. Numerical examples are in good agreement with Treloar’s classic data for rubbers and with extensive data for gellan gels up to failure.

A new elastoplastic J 2 − flow model with nonlinear combined hardening is proposed for the purpose of achieving an explicit, comprehensive simulation of pseudo-elastic SMAs over the whole range of strain up to failure. To this goal, a new, explicit methodology based on any given uniaxial data is introduced to obtain multi-axial expressions for the constitutive quantities incorporated in this model, in such a unified sense that all three successive deformation stages may be simultaneously represented, including a pseudo-elastic stage with perfect strain recovery, a hardening stage with partial strain recovery, and a softening stage up to failure. As such, any given shapes of uniaxial stress–strain curves from both loading and unloading in these three stages may be automatically reproduced from the proposed model, without involving usual complicated numerical procedures in treating nonlinear rate constitutive equations with a number of switching conditions. Numerical examples with test data are supplied to simulate a complete process including all three stages.

Numerical simulations based on Reynolds-Averaged Navier–Stokes (RANS) equations are widely used in engineering design and analysis involving turbulent flows. However, RANS simulations are known to be unreliable in many flows of engineering relevance, which is largely caused by model-form uncertainties associated with the Reynolds stresses . Recently, a machine-learning approach has been proposed to quantify the discrepancies between RANS modeled Reynolds stress and the true Reynolds stress. However, it remains a challenge to represent discrepancies in the Reynolds stress eigenvectors in machine learning due to the requirements of spatial smoothness, frame-independence, and realizability. This challenge also exists in the data-driven computational mechanics in general where quantifying the perturbation of stress tensors is needed. In this work, we propose three schemes for representing perturbations to the eigenvectors of RANS modeled Reynolds stresses: (1) discrepancy-based Euler angles , (2) direct-rotation-based Euler angles, and (3) unit quaternions . We compare these metrics by performing a priori and a posteriori tests on two canonical flows: fully developed turbulent flows in a square duct and massively separated flows over periodic hills. The results demonstrate that the direct-rotation-based Euler angles representation lacks spatial smoothness while the discrepancy-based Euler angles representation lacks frame-independence, making them unsuitable for being used in machine-learning-assisted turbulence modeling . In contrast, the representation based on unit quaternion satisfies all the requirements stated above, and thus it is an ideal choice in representing the perturbations associated with the eigenvectors of Reynolds stress tensors. This finding has clear importance for uncertainty quantification and machine learning in turbulence modeling and for data-driven computational mechanics in general.

In computational fluid dynamics simulations of industrial flows, models based on the Reynolds-averaged Navier--Stokes (RANS) equations are expected to play an important role in decades to come. However, model uncertainties are still a major obstacle for the predictive capability of RANS simulations. This review examines both the parametric and structural uncertainties in turbulence models. We review recent literature on data-free (uncertainty propagation) and data-driven (statistical inference) approaches for quantifying and reducing model uncertainties in RANS simulations. Moreover, the fundamentals of uncertainty propagation and Bayesian inference are introduced in the context of RANS model uncertainty quantification. Finally, the literature on uncertainties in scale-resolving simulations is briefly reviewed with particular emphasis on large eddy simulations., submitted to Progress in Aerospace Sciences as an invited review article

Multi-axial hyper-elastic models for large strain rubberlike elasticity are established in a broad sense free of the commonly assumed constraint of incompressibility. Results are derived directly from uniaxial stress–strain relations by means of certain explicit procedures. Novelties in four respects are incorporated in the new models: (i) constitutive parameters of direct physical meanings may be introduced to represent features of rubberlike elasticity; (ii) the usual incompressibility constraint is no longer assumed and relevant issues may be rendered irrelevant; (iii) the incompressibility case may be derived as a natural limit; and (iv) accurate agreement with benchmark data for several deformation modes may be achieved, including uniaxial extension, equi-biaxial extension as well as plane-strain extension and others.

This study is concerned with a new, explicit approach by means of which forms of the large strain elastic potential for multiaxial rubberlike elasticity may be obtained based on data for a single deformation mode. As a departure from usual studies, here for the first time errors may be estimated and rendered minimal for all possible deformation modes and, furthermore, failure behavior may be incorporated. Numerical examples presented are in accurate agreement with Treloar’s well-known data.

In contrast with the widely known elastoplastic behavior of hard solids with very small elastic deformations, we study elastoplastic behavior of soft solids dominated by very large elastic deformations. Toward this goal, thermo-coupled rate-independent and rate-dependent elastoplastic J 2 -flow models with evolving rubberlike elasticity are established for the first time in a sense of identically meeting the Clausius–Duhem inequality. Novel results are presented for coupling effects in three respects: (i) how finite strain elastic behavior may evolve with development of plastic flow; (ii) how plastic flow may be induced in a process of pure heating; and (iii) how strain rate effects may be characterized to ensure smooth transitions to the rate-independent case. It is demonstrated that complex inelastic deformation features observed in soft solids such as elastomers, including the Mullins effect, the permanent set, the induced anisotropy, the thermal recovery and the rate effect etc., may be derived as direct, natural consequences of the proposed model. In particular, explicit expressions for the constitutive functions incorporated are derivable from the uniaxial data for the purpose of achieving an explicit, exact simulation of the foregoing inelastic features, thus bypassing usual complexities both in choosing suitable forms of constitutive functions and in identifying unknown parameters in an approximate sense.

According to the well-known models for rubberlike elasticity with strainstiffening effects, the unbounded strain energy is generated with the unlimitedly growing stress when the stretch approaches certain limits. Toward a solution to this issue, an explicit approach is proposed to derive the multi-axial elastic potentials directly from the uniaxial potentials. Then, a new multi-axial potential is presented to characterize the strain-stiffening effect by prescribing suitable forms of uniaxial potentials so that the strain energy is always bounded as the stress grows to infinity. Numerical examples show good agreement with a number of test data.

In this work, a coarse-graining method previously proposed by the authors in a companion paper based on solving diffusion equations is applied to CFD–DEM simulations, where coarse graining is used to obtain solid volume fraction, particle phase velocity, and fluid–particle interaction forces. By examining the conservation requirements, the variables to solve diffusion equations for in CFD–DEM simulations are identified. The algorithm is then implemented into a CFD–DEM solver based on OpenFOAM and LAMMPS, the former being a general-purpose, three-dimensional CFD solver based on unstructured meshes. Numerical simulations are performed for a fluidized bed by using the CFD–DEM solver with the diffusion-based coarse-graining algorithm. Converged results are obtained on successively refined meshes, even for meshes with cell sizes comparable to or smaller than the particle diameter. This is a critical advantage of the proposed method over many existing coarse-graining methods, and would be particularly valuable when small cells are required in part of the CFD mesh to resolve certain flow features such as boundary layers in wall bounded flows and shear layers in jets and wakes. Moreover, we demonstrate that the overhead computational costs incurred by the proposed coarse-graining procedure are a small portion of the total computational costs in typical CFD–DEM simulations as long as the number of particles per cell is reasonably large, although admittedly the computational overhead of the coarse-graining procedure often exceeds that of the CFD solver. Other advantages of the diffusion-based algorithm include more robust and physically realistic results, flexibility and easy implementation in almost any CFD solvers, and clear physical interpretation of the computational parameter needed in the algorithm. In summary, the diffusion-based method is a theoretically elegant and practically viable option for practical CFD–DEM simulations.

Proper quantification and propagation of uncertainties in computational simulations are of critical importance. This issue is especially challenging for computational fluid dynamics (CFD) applications. A particular obstacle for uncertainty quantifications in CFD problems is the large model discrepancies associated with the CFD models used for uncertainty propagation. Neglecting or improperly representing the model discrepancies leads to inaccurate and distorted uncertainty distribution for the quantities of interest (QoI). High-fidelity models, being accurate yet expensive, can accommodate only a small ensemble of simulations and thus lead to large interpolation errors and/or sampling errors; low-fidelity models can propagate a large ensemble, but can introduce large modeling errors. In this work, we propose a multimodel strategy to account for the influences of model discrepancies in uncertainty propagation and to reduce their impact on the predictions. Specifically, we take advantage of CFD models of multiple fidelities to estimate the model discrepancies associated with the lower-fidelity model in the parameter space. A Gaussian process (GP) is adopted to construct the model discrepancy function, and a Bayesian approach is used to infer the discrepancies and corresponding uncertainties in the regions of the parameter space where the high-fidelity simulations are not performed. Several examples of relevance to CFD applications are performed to demonstrate the merits of the proposed strategy. Simulation results suggest that, by combining low- and high-fidelity models, the proposed approach produces better results than what either model can achieve individually.

Rate-independent finite elastoplastic equations with thermo-coupled effects are proposed to bypass the yield condition and loading–unloading conditions. These new equations are shown to be more realistic and of much simpler structure than classical equations. In a sense of thermodynamic consistency the strength property of elastoplastic solids is then studied from the standpoint of stress-bearing capacity. It is shown that asymptotic loss of the strength may be derived directly from elastoplastic equations, thus leading to the finding of a noticeable phenomenon, namely, asymptotic vanishing of the stress concurrent with developing elastoplastic flow. It is indicated that this finding may suggest a natural constitutive characterization of fatigue, fracture and failure as certain limiting cases of elastoplastic behavior. Simple models with asymptotic loss of the strength are constructed as examples of potential practical applications.

Multi-axial elastic potentials for isotropic, incompressible elastomeric solids are constructed based solely on uniaxial potentials by means of direct, explicit procedures. Results are presented for the purpose of meeting the following three requirements: (i) the strain-stiffening effect is represented with rapidly growing stress at certain strain limits, (ii) the strain energy never grows to infinity but is always bounded, and (iii) the stress is also bounded and asymptotically tends to vanish with increasing strain up to failure. As such, a realistic simulation of rubberlike elasticity with the strain-stiffening effect up to failure is proposed for the first time. Numerical examples show good agreement with a number of test data.

An explicit, exact approach is proposed to obtain multi-axial elastic potentials for isotropic rubber-like materials undergoing large incompressible deformations. By means of two direct, explicit procedures, this approach reduces the problem of determining multi-axial potentials to that of determining one-dimensional elastic potentials. To this end, two one-dimensional potentials for uniaxial case and simple shear case are respectively determined via spline interpolation and, then, the two potentials are extended to generate a multi-axial elastic potential using a novel method based on certain logarithmic invariants. Eventually, each of the multi-axial potentials will exactly match the finite strain data from four benchmark tests.

An explicit approach is proposed to determine elastic potentials of elastomers for the purpose of eliminating the long-standing uncertainty that the applicability may not be assessed for all deformation modes. Toward this goal, a new experimental scheme for testing nonlinear elastic characteristics of elastomers is first designed for obtaining suitable test data, and then, multi-axial elastic potentials are directly constructed via explicit procedures based on these data. For the first time, the error estimate may be established and minimized errors may be achieved for all possible deformation modes by combining Hermite interpolation with Chebyshev’s best approximation. It is demonstrated that each potential thus obtained may with sufficient accuracy represent nonlinearly elastic properties for all deformation modes. It is shown that even the simplest case of the proposed potential, derived from the uniaxial mode alone, may achieve good agreement with a number of test data.

Toward a straightforward approach to modeling SMA pseudoelastic behavior, a new finite J 2 -flow elastoplastic model with nonlinear combined hardening is proposed by coupling the size change of the yield surface with the moving of the yield surface center. A departure from usual approaches is that any given shapes of hysteresis loops may be automatically generated via explicit procedures in the framework of classical J 2 elastoplasticity with no reference to any phase variables. Numerical examples are presented to show good accord with test data for Ti–Ni alloys.

By virtue of the rational interpolation procedure and logarithmic strain, a direct approach is proposed to obtain elastic potentials that exactly match uniaxial data and shear data for elastomers. This approach reduces the determination of multiaxial elastic potentials to that of two one-dimensional potentials, thus bypassing usual cumbersome procedures of identifying a number of unknown parameters. Predictions of the suggested potential are derived for a general biaxial stretch test and compared with the classical data given by Rivlin and Saunders (Rivlin, R. S. and Saunders, D. W. Large elastic deformation of isotropic materials. VII: experiments on the deformation of rubber. Phill. Trans. Royal Soc. London A, 243, 251–288 (1951)). Good agreement is achieved with these extensive data.

This paper presents a flameless catalytic combustion-based thermoelectric power generator that uses commercial thermoelectric modules. The structure of the thermoelectric generator (TEG) is introduced and the power performance is measured based on a designed circuit system. The open circuit voltage of the TEG is about 7.3 V. The maximum power output can reach up to 6.5 W when the load resistance matches the TEG internal resistance. However, the system output is sensitive to load variation. To improve this characteristic, maximum power point tracking technique is used and results in an open circuit voltage of 13.8 V. The improved characteristic makes the TEG system a good charger to keep the lead acid battery fully charged so as to meet the needs of electronic instruments on gas pipelines. In addition, the combustion features have been investigated based on the temperature measurement. Test results show that the uniformity of combustion heat transfer process and the combustion chamber structure play important roles in improving system power output. It can get an optimized TEG system (maximum power output: 8.3 W) by uniformly filling a thermal insulation material (asbestos) to avoid a non-uniform combustion heat transfer process.

Thermoelectric devices can provide clean energy conversion and are environmentally friendly. However, few studies have been conducted to understand the dynamic characteristics of (thermoelectric generators) TEGs. The stability of output performance plays a very important role in TEG. Based on the thermoelectric effect and heat transfer theory, a dynamic model for waste heat recovery in a general TEG was developed to asses the influence of heat reservoir and heat sink. The validity of the theoretical model was demonstrated by experiments and it was found that the model prediction agrees well with the experimental results. The dynamic response characteristics, such as hot and cold semiconductor surface temperatures, maximum output power and system efficiency, were studied using this dynamic model. The results showed that enhancing heat dissipation on cold side is the crucial to improve the output performance of TEG, and that the fluctuation of hot reservoir leads to rapid change of output power generated by TEG, which is dangerous to electric devices. The dynamic model proposed in the present study can be used in the design and operation analysis of TEGs.

The main objective of this study is to demonstrate that recoverable finite elastoplastic flows are derivable from classical elastoplasticity models and to further establish combined hardening finite J 2 -flow models of this property. With such models, it is shown that every axial strain of a uniaxially loaded bar well beyond the initial yielding is recoverable upon removal of applied load, thus producing a hysteresis loop above the strain axis. It is found that such hysteresis loops of any given shape may be generated by specifying suitable hardening functions. In particular, hardening functions are found to generate flag-like loops associated just with pseudoelastic hysteresis effects of shape memory alloys. This suggests that these effects may be characterized by classical elastoplasticity models from a direct phenomenological standpoint. Explicit, direct results are derived for a general case of multiaxial deformations in a sense of thermodynamic consistency.

With the ever-increasing use of Reynolds-Averaged Navier--Stokes (RANS) simulations in mission-critical applications, the quantification of model-form uncertainty in RANS models has attracted attention in the turbulence modeling community. Recently, a physics-based, nonparametric approach for quantifying model-form uncertainty in RANS simulations has been proposed, where Reynolds stresses are projected to physically meaningful dimensions and perturbations are introduced only in the physically realizable limits. However, a challenge associated with this approach is to assess the amount of information introduced in the prior distribution and to avoid imposing unwarranted constraints. In this work we propose a random matrix approach for quantifying model-form uncertainties in RANS simulations with the realizability of the Reynolds stress guaranteed. Furthermore, the maximum entropy principle is used to identify the probability distribution that satisfies the constraints from available information but without introducing artificial constraints. We demonstrate that the proposed approach is able to ensure the realizability of the Reynolds stress, albeit in a different manner from the physics-based approach. Monte Carlo sampling of the obtained probability distribution is achieved by using polynomial chaos expansion to map independent Gaussian random fields to the Reynolds stress random field with the marginal distributions and correlation structures as specified. Numerical simulations on a typical flow with separation have shown physically reasonable results, which verifies the proposed approach. Therefore, the proposed method is a promising alternative to the physics-based approach for model-form uncertainty quantification of RANS simulations. The method explored in this work is general and can be extended to other complex physical systems in applied mechanics and engineering., 42 pages, 10 figures

The growth of computational resources in the past decades has expanded the application of Computational Fluid Dynamics (CFD) from the traditional fields of aerodynamics and hydrodynamics to a number of new areas. Examples range from the heat and fluid flows in nuclear reactor vessels and in data centers to the turbulence flows through wind turbine farms and coastal vegetation plants. However, in these new applications complex structures are often exist (e.g., rod bundles in reactor vessels and turbines in wind farms), which makes fully resolved, first-principle based CFD modeling prohibitively expensive. This obstacle seriously impairs the predictive capability of CFD models in these applications. On the other hand, a limited amount of measurement data is often available in the systems in the above-mentioned applications. In this work we propose a data-driven, physics-based approach to perform full field inversion on the effects of the complex structures on the flow. This is achieved by assimilating observation data and numerical model prediction in an iterative Ensemble Kalman method. Based on the inversion results, the velocity and turbulence of the flow field can be obtained. A major novelty of the present contribution is the non-parametric, full field inversion approach adopted, which is in contrast to the inference of coefficient in the ad hoc models often practiced in previous works. The merits of the proposed approach are demonstrated on the flow past a porous disk by using both synthetic data and real experimental measurements. The spatially varying drag forces of the porous disk on the flow are inferred. The proposed approach has the potential to be used in the monitoring of complex system in the above mentioned applications.