First-order Stress Research Articles

Optimising Physics-Informed Neural Network Solvers for Turbulence Modelling: A Study on Solver Constraints Against a Data-Driven Approach

Physics-informed neural networks (PINNs) have emerged as a promising approach for simulating nonlinear physical systems, particularly in the field of fluid dynamics and turbulence modelling. Traditional turbulence models often rely on simplifying assumptions or closed numerical models, which simplify the flow, leading to inaccurate flow predictions or long solve times. This study examines solver constraints in a PINNs solver, aiming to generate an understanding of an optimal PINNs solver with reduced constraints compared with the numerically closed models used in traditional computational fluid dynamics (CFD). PINNs were implemented in a periodic hill flow case and compared with a simple data-driven approach to neural network modelling to show the limitations of a data-driven model on a small dataset (as is common in engineering design). A standard full equation PINNs model with predicted first-order stress terms was compared against reduced-boundary models and reduced-order models, with different levels of assumptions made about the flow to monitor the effect on the flow field predictions. The results in all cases showed good agreement against direct numerical simulation (DNS) data, with only boundary conditions provided for training as in numerical modelling. The efficacy of reduced-order models was shown using a continuity only model to accurately predict the flow fields within 0.147 and 2.6 percentage errors for streamwise and transverse velocities, respectively, and a modified mixing length model was used to show the effect of poor assumptions on the model, including poor convergence at the flow boundaries, despite a reduced solve time compared with a numerically closed equation set. The results agree with contemporary literature, indicating that physics-informed neural networks are a significant improvement in solve time compared with a data-driven approach, with a novel proposition of numerically derived unclosed equation sets being a good representation of a turbulent system. In conclusion, it is shown that numerically unclosed systems can be efficiently solved using reduced-order equation sets, potentially leading to a reduced compute requirement compared with traditional solver methods.

Improved mesh-free SPH approach for loose top coal caving modeling

This study presents an innovative model in computational geotechnical engineering by improving the Smoothed Particle Hydrodynamics (SPH) method for simulating loose particle dynamics in coal caving processes. The improved model integrates an elastic-perfectly plastic constitutive model with the Drucker-Prager yield criterion and includes several improvements aimed at boosting accuracy, stability, and efficiency. These improvements include gravity loading coupled with particle damping, first-order stress field smoothing, and kernel gradient correction. A series of numerical experiments validates the effectiveness of the improved SPH model, demonstrating its capability to predict large deformations and track the evolution of the coal-rock interface in coal caving processes. Furthermore, the study analyzes the model's sensitivity to material parameters such as the angle of friction and material density, which aids in configuring the model for distinct coal mining situations. Results show that the non-cohesive elastic-perfectly plastic constitutive model can effectively simulate the flow behavior of granular particles, and the landslide simulation results are in good agreement with the experiments. The improved SPH algorithm with stress smoothing technique solves the problem of numerical noise, and the “double peak” stress distribution around the coal outlet is identified. The established SPH model offers an effective tool for understanding dynamics behaviors of loose top coal. Significantly, the model requires only five material parameters, which can be identified through standard experiments, avoiding the typically arduous process of parameter selection or calibration commonly existing in Discrete Element Method simulations.

The 3D stress field of Nordland, northern Norway – insights from numerical modelling

Abstract The Nordland area in northern Norway is the seismically most active area on mainland Fennoscandia. It exhibits patterns of coastal extension, which contrasts with the first-order regional stress pattern that reflects compression aligned with the North Atlantic ridge push. The regional stress field has been considered to emanate from the interaction of ridge push and glacial isostatic adjustment; while the local stress pattern can be additionally influenced by gravitational, topographic stresses, as well as the flexural effects of erosion and sediment deposition. We employ finite element numerical models at a crustal scale to study the 3D stress field, using existing geometric constraints from previous geophysical studies. Internal body forces, induced by variations in density, topography or Moho depth, already yield significant deviatoric stresses. In the models tested, these can strongly influence the near-surface stress regime, in particular for the continental margin setting we are investigating. In addition, redistribution of rock mass, which occurred mainly under Pleistocene glaciation, can modify the stress field significantly on a semi-regional scale. We consider this process to be the main driver for the coastal extension, in particular in areas where erosion has been high.

Variations in the present-day stress field and its implications for hydrocarbon development in the Hassi Terfa field, Hassi Messaoud, Algeria

Understanding the contemporary stress field and its variations is important in hydrocarbon production. These fluctuations are significant for borehole stability and trajectory, hydraulic stimulation, fault reactivation, and optimal well completion. In this study, we provide the first geomechanical model to determine the orientation and magnitude of the present-day stress of the Hassi Terfa field using geophysical logs to understand the origin of the spatial variation of the stress state. The breakouts from acoustic image logs provide an average maximum horizontal stress (SHmax) orientation of N115°, consistent with the Africa–Eurasia plate motion, which imposes the largest first-order stress source. However, our analysis demonstrates local stress perturbations near faults. This may mean that the stress field in the Hassi Terfa is superimposed by a third-order source of stress (<100 km). Our model demonstrates variations in stress magnitude with depth interpreted to be lithology-dependent, which act as a natural barrier against fracture propagation. We propose a new approach to quantify these perturbations via the stress regime index R′, using the magnitude of the principal stress components. The results demonstrate different layer-to-layer faulting regimes. These perturbations will limit the vertical propagation of hydraulic fracturing to the overlying formation, which enables it to access a larger reservoir volume and improve production. In general, a strike–slip regime with a slight normal slip component is interpreted for the study area. We develop the first-ever quantitative stress map using the relative principal stress magnitude parameter (Aϕ) to present the lateral variation of the stress regime throughout the Saharan platform. Results demonstrate a consistent strike–slip to normal/strike–slip regime with a uniform SHmax orientation. This map will help operators achieve optimal well direction and optimize hydraulic stimulation. Our results have considerable implications for hydrocarbon production in the central and southeastern Saharan platform.

A refined catalogue of focal mechanisms for the Intra‑Carpathian region of Romania: implications for the stress field and seismogenic features assessment

We present a refined and the most complete catalogue of the focal mechanisms for the Intra-Carpathian region of Romania. It contains the high-quality solutions computed for 1217 earthquakes recorded between 1909 and 2018. Primary data gathered from the original seismograms and seismic bulletins have been used to compute the source parameters and focal mechanisms solutions. The focal mechanisms have been obtained using the HASH method by the polarities and S/P amplitudes ratios inversion. Our catalogue provides data necessary for the investigation of the contemporary stress field at different scales with high spatial and temporal resolution. We determined the stress field characteristics through formal inversion of focal mechanisms and also computed the reactivation potential of the active fault systems using the Win‑Tensor program. The stress field is heterogeneous, with SHmax significantly deviating from the first-order stress field direction and also with strong local variations in the stress regime.

Stress-based dimensional reduction and dual-mixed hp finite elements for elastic plates

Starting from the linearized weak forms of the kinematic equation and the angular momentum balance equation of three-dimensional non-linear elasticity, a stressbased dimensional reduction procedure is presented for elastic plates. After expanding the three-dimensional non-symmetric stress tensor into power series with respect to the thickness coordinate, the translational equilibrium equations, written in terms of the expanded stress coefficients, are satisfied by introducing first-order stress functions. The symmetry of the stress field is satisfied in a weak sense by applying the material rotations as Lagrangian multipliers. The seven-field plate model developed in this way employs unmodified three-dimensional strain-stress relations. On the basis of the dimensionally reduced plate model derived, a new dual-mixed plate bending finite element model is developed and presented. The numerical performance of the hp-version plate elements is investigated through the solutions of standard plate bending problems. It is shown that the modeling error of the stress-based plate model in the energy norm is better than that of the displacement-based Kirchhoff- and Reissner-Mindlin plate models. The numerical solutions and their comparisons to reference solutions indicate that the dual-mixed hp elements are free from locking problems, in either the energy norm or the stress computations, both for h- and p-extensions, and the results obtained for the stresses are accurate and reliable even for extremely thin plates.

Elastic wave pre-stack reverse-time migration based on the second-order P- and S-wave decoupling equation

Abstract The problems of large computation, large storage and low-frequency noises have been the key factors limiting the development of elastic wave pre-stack reverse-time migration (RTM). Most research has centered on the wave equation of first-order velocity stress and using Helmholtz separation to get pure primary and secondary waves. However, the cost is enormous and the amplitude and phase of wavefields will be changed. So, we use the second-order P- and S-wave decoupling equations to construct seismic wavefields that reduce the number of variables, and the storage space required for boundary wavefields is reduced by ∼78%. Then we bring the boundary-saving approach into our study during the wavefield extrapolation process so that we can reconstruct the source wavefields completely. Finally, for the wave equations of second-order displacement used in this study, the Poynting vector expressed by the traditional method was not applicable. Therefore, we derived the formula of the energy flux density vector represented by displacements and used it in directional traveling wave decomposition to suppress low-frequency noise in the imaging profile. Subsequently, based on the inner product imaging conditions, the RTM of the 2-D salt dome model was calculated. The results illustrate the effectiveness and practicability of the proposed solution.

Stress rotation – impact and interaction of rock stiffness and faults

Abstract. It has been assumed that the orientation of the maximum horizontal compressive stress (SHmax) in the upper crust is governed on a regional scale by the same forces that drive plate motion. However, several regions are identified where stress orientation deviates from the expected orientation due to plate boundary forces (first-order stress sources), or the plate wide pattern. In some of these regions, a gradual rotation of the SHmax orientation has been observed. Several second- and third-order stress sources have been identified in the past, which may explain stress rotation in the upper crust. For example, lateral heterogeneities in the crust, such as density and petrophysical properties, and discontinuities, such as faults, are identified as potential candidates to cause lateral stress rotations. To investigate several of these candidates, generic geomechanical numerical models are set up with up to five different units, oriented by an angle of 60∘ to the direction of shortening. These units have variable (elastic) material properties, such as Young's modulus, Poisson's ratio and density. In addition, the units can be separated by contact surfaces that allow them to slide along these vertical faults, depending on a chosen coefficient of friction. The model results indicate that a density contrast or the variation of Poisson's ratio alone hardly rotates the horizontal stress (≦17∘). Conversely, a contrast of Young's modulus allows significant stress rotations of up to 78∘, even beyond the vicinity of the material transition (&gt;10 km). Stress rotation clearly decreases for the same stiffness contrast, when the units are separated by low-friction discontinuities (only 19∘ in contrast to 78∘). Low-friction discontinuities in homogeneous models do not change the stress pattern at all away from the fault (&gt;10 km); the stress pattern is nearly identical to a model without any active faults. This indicates that material contrasts are capable of producing significant stress rotation for larger areas in the crust. Active faults that separate such material contrasts have the opposite effect – they tend to compensate for stress rotations.

Stress rotation – impact and interaction of rock stiffness and faults

Abstract. It has been assumed that the orientation of the maximum horizontal compressive stress ( SHmax ) in the upper crust is governed on a regional scale by the same forces that drive plate motion. However, several regions are identified where stress orientation deviates from the expected orientation due to plate boundary forces (first-order stress sources), or the plate wide pattern. In some of these regions, a gradual rotation of the SHmax orientation has been observed. Several second- and third-order stress sources have been identified in the past, which may explain stress rotation in the upper crust. For example, lateral heterogeneities in the crust, such as density and petrophysical properties, and discontinuities, such as faults, are identified as potential candidates to cause lateral stress rotations. To investigate several of these candidates, generic geomechanical numerical models are set up with up to five different units, oriented by an angle of 60 ∘ to the direction of shortening. These units have variable (elastic) material properties, such as Young's modulus, Poisson's ratio and density. In addition, the units can be separated by contact surfaces that allow them to slide along these vertical faults, depending on a chosen coefficient of friction. The model results indicate that a density contrast or the variation of Poisson's ratio alone hardly rotates the horizontal stress ( ≦17 ∘ ). Conversely, a contrast of Young's modulus allows significant stress rotations of up to 78 ∘ , even beyond the vicinity of the material transition ( >10 km ). Stress rotation clearly decreases for the same stiffness contrast, when the units are separated by low-friction discontinuities (only 19 ∘ in contrast to 78 ∘ ). Low-friction discontinuities in homogeneous models do not change the stress pattern at all away from the fault ( >10 km ); the stress pattern is nearly identical to a model without any active faults. This indicates that material contrasts are capable of producing significant stress rotation for larger areas in the crust. Active faults that separate such material contrasts have the opposite effect – they tend to compensate for stress rotations.

Analytical solution of stress and displacement for a circular underwater shallow tunnel based on a unified stress function

A unified stress function is represented in a Fourier series to capture the complex stress state of an underwater shallow tunnel perimeter. The first-order unified stress function is employed as stress boundary condition of the tunnel perimeter (SBCTP) to describe the non-symmetrical stress with respect to the horizontal and vertical centerlines of the tunnel. The mechanical analysis model of an underwater shallow tunnel is established. Combining the complex variable theory, the elastic analytical solution of the stress and displacement are obtained. The rationality of the elastic solution is verified by comparing the stress fields between the proposed method and the finite element method (FEM). Based on the obtained elastic solution, the fractional viscoelastic solution is determined by the correspondence principle. The time-dependent ground displacement are investigated considering the effect of stress release. A series of parameter analyses are carried out to discuss the ground vertical and horizontal stress distributions under the influences of SBCTPs.

Rheological, functional and thermal properties of the blend system of canary seed starch-wheat starch gels

In this paper, some functional, thermal, textural, and rheological properties of the blends of two canary seed starches (CDC Maria and C05041 varieties) and wheat starch (WS) were studied to assess their suitability as an alternative for chemically modified starches. The blends of CDC Maria starch (CSS1-WS) and C05041 starch (CSS2-WS) were prepared in the ratio of 25:75, 50:50, and 75:25. The blend with the highest proportion of CSS2 (CSS2-75) represented the highest water and oil absorption capacity, more temperature dependency, lesser syneresis, and thus the highest freeze–thaw stability. The results of rheological properties revealed that the Power-law, Casson, and first-order stress decay models were the best ones for describing the time-independent and time-dependent behaviors of the selected starches, and their blends, respectively. The gained output from the Power-law model displayed that the pseudoplasticity of all starch gels reduced by increasing the CSS portions into the mixture. Increasing the ratio of CSSs in the mixture from 25 to 75% caused an appreciable increase in both yield stress (12.52–20.13%) and plastic viscosity (16.47–33.73%) values. The extent of thixotropy for the CSS1-WS blend decreased by increasing the portion of CSS1 in the mixture, opposite of the effect of raising the portion of CSS2 in the mixture.

Effect of the initial ECAP passes on crystal texture and residual stresses of 5083 aluminum alloy

To produce a highly refined microstructure, several metals or alloys have been processed via equal-channel angular pressing (ECAP). In this work, the mechanical and microstructural changes of the 5083 aluminum alloy in H11 condition after processed by two ECAP passes were investigated. An ECAP H13 steel die with an inner angle (α) of 120° and outer curvature (β) of 20° was used. The microstructural changes were associated with the loss of texture symmetry. The morphologies of the Mg2Si and α-Al(Mn,Fe)Si precipitates for the sample at the initial condition were similar to those subjected to two ECAP passes. The peak broadening measured by X-ray diffraction revealed an increment of both grain refinement and microstrain. After the second extrusion pass, the hardness increased by 62% compared with the initial condition. Moreover, the heterogeneous hardness behavior was compatible with a highly localized dislocation density. After two ECAP passes, shear parallel bands were observed to be at nearly 45° to the extrusion direction. The evaluation of first-order residual stress as a function of the depth of the analyzed sample displayed compressive or tensile values, depending on the measured face. With the plastic deformation applied, the first and second-order residual stresses exhibited significant increment. Williamson-Hall plots showed positive slopes, indicating an increment in the microstrain.

A filter-based level set topology optimization method using a 62-line MATLAB code

The present paper proposes a fast and easy to implement level set topology optimization method that is able to adjust the complexity of resulting configurations. While the existing methods solve a large system of linear equations, the proposed method applies a density filter to the level set function in order to smoothen the optimized configurations. Considering the proposed method, the minimum total potential energy problem and optimum design problem of compliant mechanism will be studied in the present study. In topology optimization methods, the topological derivative determines the criteria of removing/adding material from/to the domain of definition of the problem. We utilize the definition of topological derivative to measure the sensitivity of a given objective function when an infinite small hole is inserted at an arbitrary location of the domain. It is shown that a first-order stress field appears in the topological derivative. Moreover, a comprehensive Airy stress function is proposed in order to solve the boundary value problem that governs the appeared first-order stress field. Finally, a simple MATLAB implementation is provided to confirm the effectiveness and utility of the proposed method.

A Cook-féle membrán feladat megoldása p-verziós végeselem-modellekkel

Ez a cikk egy szakirodalomból jól ismert sík-rugalmasságtani peremérték-feladat, a Cook-féle membrán probléma numerikus megoldásán keresztül veti össze az elsőrendű feszültségfüggvényeken és forgásokon alapuló, p-verziós dual-mixed végeselem-modellt és a klasszikus, elmozdulásmezőn alapuló p-verziós végeselem-modellt.

Hypersingular boundary integral equations for plane orthotropic elasticity in terms of first-order stress functions

This paper is intended to present an implementation of the hypersingular boundary integral equations in terms of first-order stress functions for stress computations in plane orthotropic elasticity. In general, the traditional computational technique of the boundary element method used for computing the stress distribution on the boundary and close to it is not as accurate as it should be. In contrast, the accuracy of stress computations on the boundary is greatly increased by applying the hypersingular integral equations. Contrary to the method in which the solution is based on an approximation of displacement field, here the first-order stress functions and the rigid body rotation are the fundamental variables. An advantage of this approach is that the stress components can be obtained directly from the stress functions, there is, therefore, no need for Hooke's law, which should be used when they are computed from displacements. In addition, the computational work can be reduced when the stress distribution is computed at an arbitrary point on the boundary. The numerical examples presented prove the efficiency of this technique.

Equilibrated stress space for nonlinear dimensionally reduced shell models in terms of first-order stress functions

Considering the power series expansion of the three-dimensional variables with respect to the shell thickness coordinate, an equilibrated stress space for the first Piola-Kirchhoff stress vectors is derived in convective curvilinear coordinate system. The infinite series of the two-dimensional translational equilibrium equations are satisfied by introducing two first-order stress function vectors expanded into power series. For the important case of thin shells, the infinite number of two-dimensional equilibrium equations is truncated to obtain a ‘first-order’ model, where the equilibrated stress-space requires three vectorial stress function coefficients only. The formulation presented for thin shells is compared to the nonlinear equilibrium equations of the classical shell theories, written in terms of the first Piola-Kirchhoff stress resultants and stress couples and satisfied by the introduction of three first-order stress function vectors.

Method of manufactured solutions code verification of elastostatic solid mechanics problems in a commercial finite element solver

Much progress has been made in advancing and standardizing verification, validation, and uncertainty quantification practices for computational modeling in recent decades. However, examples of rigorous code verification for solid mechanics problems in literature remain scarce, particularly for commercial software and for the non-trivial large-deformation analyses and nonlinear materials typically needed to simulate medical devices. Here, we apply the method of manufactured solutions (MMS) to verify a commercial finite element code for elastostatic solid mechanics analyses using linear-elastic, hyperelastic (neo-Hookean), and quasi-hyperelastic (Hencky) constitutive models. Analytical source terms are generated using Python/SymPy and are implemented in ABAQUS/Standard without modification to solver source code. Source terms for the three constitutive models are found to vary nearly six orders of magnitude in the number of mathematical operations they contain. Refinement studies reveal second-order displacement and first-order (or higher) stress and strain convergence in response to mesh refinement for all constitutive models and first-order displacement convergence in response to pseudo-time increment refinement for the Hencky-elastic case. We also investigate the sensitivity of convergence order to quantitatively minor changes to the underlying mathematical model using an exploratory case. Code used to generate the MMS source terms and simulation input files are provided as Supplemental Material.

Rheological and interfacial properties of basil seed gum modified with octenyl succinic anhydride

This study aimed to evaluate rheological and interfacial properties of basil seed gum (BSG) esterified with octenyl succinic anhydride (OSA) at different OSA:BSG weight ratios (WRs) of 0, 0.01, and 0.03. The amounts of added OSA were analyzed by high performance liquid chromatography (HPLC) and ion mobility spectroscopy (IMS). The high correlation coefficient (R2 = 0.998) between the results of HPLC (0%, 0.277%, and 1.01%) and IMS (0%, 0.31%, and 0.97%), obtained at different WRs, indicated that IMS can be considered as an alternative analytical technique for HPLC to determine the extent of modification. The entropy (image information content) of scanning electron micrographs of lyophilized BSG was decreased after modification and attributed to relative disappearance of spherical particles and formation of a structure with higher integrity. A decrease in interfacial tension, an increase in contact angle and molecular weight, and more negative values of zeta-potential were recorded after modification. All dispersions showed shear-thinning behavior with an increase in apparent viscosity after modification. The first-order stress decay with a non-zero equilibrium stress was better than other models for predicting the thixotropic properties. The dilute solution properties were better fitted with slope-based models than intercept-based models. A dominant elastic behavior was observed in BSG dispersions and corresponding BSG-stabilized emulsions and improved after modification. The OSA-modified BSG exhibited an improvement in the emulsifying and foaming capacities and colloidal stability over time. Emulsions prepared with modified gums showed a smaller droplet size. OSA-modified BSG might be a good candidate for improving the long-term stability of emulsions.

Irregular surface seismic forward modeling by a body-fitted rotated–staggered-grid finite-difference method

Finite-difference (FD) methods are widely used in seismic forward modeling owing to their computational efficiency but are not readily applicable to irregular topographies. Thus, several FD methods based on the transformation to curvilinear coordinates using body-fitted grids have been proposed, e.g., stand staggered grid (SSG) with interpolation, nonstaggered grid, rotated staggered grid (RSG), and fully staggered. The FD based on the RSG is somewhat superior to others because it satisfies the spatial distribution of the wave equation without additional memory and computational requirements; furthermore, it is simpler to implement. We use the RSG FD method to transform the firstorder stress–velocity equation in the curvilinear coordinates system and introduce the highprecision adaptive, unilateral mimetic finite-difference (UMFD) method to process the freeboundary conditions of an irregular surface. The numerical results suggest that the precision of the solution is higher than that of the vacuum formalism. When the minimum wavelength is low, UMFD avoids the surface wave dispersion. We compare FD methods based on RSG, SEM, and nonstaggered grid and infer that all simulation results are consistent but the computational efficiency of the RSG FD method is higher than the rest.

Reissner stationary variational principle for nonlocal strain gradient theory of elasticity

Abstract The general form of Reissner stationary variational principle is established in the framework of the nonlocal strain gradient theory of elasticity. Including two size-dependent characteristic parameters, the nonlocal strain gradient elasticity theory can demonstrate the significance of the strain gradient as well as the nonlocal elastic stress field. Based on the Reissner functional, the governing differential and boundary conditions of dynamic equilibrium and differential constitutive equations of the classical and first-order nonlocal stress tensor are derived in the most general form. Additionally, the boundary congruence conditions are formulated and discussed for the nonlocal strain gradient theory. To exhibit the application value of Reissner variational principle, it is employed to examine the nonlinear vibrations of size-dependent Bernoulli-Euler and Timoshenko beams. In the case of immovable boundary conditions, employing the weighted residual Galerkin method, the homotopy analysis method is also utilized to determine the closed form analytical solutions of the geometrically nonlinear vibration equations. Consequently, the analytical expressions for the nonlinear natural frequencies of Bernoulli-Euler and Timoshenko nonlocal strain gradient beams are derived.