Saint Venant Research Articles

Anomalous frozen evanescent phonons

Evanescent Bloch waves are eigensolutions of spatially periodic problems for complex-valued wavenumbers at finite frequencies, corresponding to solutions that oscillate in time and space and that exponentially decay in space. Such evanescent waves are ubiquitous in optics, plasmonics, elasticity, and acoustics. In the limit of zero frequency, the wave “freezes” in time. We introduce frozen evanescent waves as the eigensolutions of the Bloch periodic problem at zero eigenfrequency. Elastic waves, i.e., phonons, in metamaterials serve as an example. We show that, in the complex plane, the Cauchy-Riemann equations for analytical functions connect the minima of the phonon band structure to frozen evanescent phonons. Their exponential decay length becomes unusually large if a minimum in the band structure tends to zero and thereby approaches a soft mode. This connection between unusual static and dynamic behaviors allows to engineer large characteristic decay lengths in static elasticity. For finite-size samples, the static solutions for given boundary conditions are linear combinations of frozen evanescent phonons, leading to interference effects. Theory and experiment are in excellent agreement. Anomalous behavior includes the violation of Saint Venant’s principle, which means that large decay-length frozen evanescent phonons can potentially be applied in terms of remote mechanical sensing.

Existence and Stability of Nonmonotone Hydraulic Shocks for the Saint Venant Equations of Inclined Thin-Film Flow

Extending the work of Yang–Zumbrun for the hydrodynamically stable case of Froude number F<2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F<2$$\\end{document}, we categorize completely the existence and convective stability of hydraulic shock profiles of the Saint Venant equations of inclined thin film flow. Moreover, we confirm by numerical experiment that asymptotic dynamics for general Riemann data is given in the hydrodynamic instability regime by either stable hydraulic shock waves, or a pattern consisting of an invading roll wave front separated by a finite terminating Lax shock from a constant state at plus infinity. Notably, profiles, and existence and stability diagrams, are all rigorously obtained by mathematical analysis and explicit calculation.

Numerical verification of torsion-bar models based on the modified couple stress theory

This study verifies the suitability of the classical Saint–Venant’s displacement field, originally intended for macroscale torsional behavior, to microscale torsion-bar models within the modified couple stress theory framework. Finite element analysis was conducted for bars with circular and rectangular cross-sections, and the results align closely with analytic solutions, which is confirmed by the energy ratio analysis. These results affirm the appropriateness of the displacement field for microscale torsion bars. Additionally, warping effects are observed to be minimal, even at the microscale. These findings validate torsion experiments for determining a material’s length scale parameter, advancing the understanding of microscale torsion.

Flood Simulation in the Complex River Basin Affected by Hydraulic Structures Using a Coupled Hydrological and Hydrodynamic Model

Due to the complexity of terrain and climate in the mountain–plain transition zone, it is difficult to simulate and forecast the flow discharge of river basins accurately. The poor regularity of the river thus leads to uncertain flood control scheduling. Meanwhile, reservoirs and flood detention areas are constructed to store and divert water when severe floods threaten the safety of the basin. In order to improve the accuracy of flood forecasts and the effectiveness of flood control, a hydrological and 1D/2D hydrodynamic coupling model was developed to enable the joint computation of multiple objects, including mountainous streams, plains river networks, hydraulic control structures, and flood detention areas. For the hydrological component, the Xin’anjiang model with the Muskingum module is employed to simulate mountainous flow discharge. For the hydrodynamic component, the Saint–Venant equations and shallow water equations are applied to estimate flood processes in rivers and on land surfaces, respectively. The Dongtiaoxi River Basin in Zhejiang Province, China, serves as the case study, where river flow is influenced by both upstream mountainous floods and downstream backwater effects. Using the integrated model, flood routing and scheduling are simulated and visualized. Both the Xin’anjiang model and the 1D hydrodynamic model demonstrate over 80% acceptability in calibration and validation, confirming their robustness and reliability. Meanwhile, inundation in flood detention areas can be effectively estimated by coupling the 1D and 2D hydrodynamic models with a flood diversion scheduling model. The coupled model proves capable of simulating flood routing in complex river basins that include mountains, plains, and hydraulic control structures, accounting for the interactions between hydrological elements. These findings provide a new perspective on flood simulation in other similarly complex river basins.

NUMERICAL COMPUTATION OF ONE- AND TWO-LAYER SHALLOW FLOW MODEL

In this research, we study a proficient computational model designed to simulate shallow flows involving one- and two-layer shallow flow. This numerical model is built upon the Saint Venant equations, which are widely used in hydraulics to depict the behavior of shallow water flow. The numerical scheme used here is constructed based on the conventional leapfrog technique implemented on a staggered grid framework, referred to as MCS. The primary objective of this research is to re-examine and implement the MCS in accurately modelling the free surface and interface waves produced by different flows passing through irregular geometries. Unlike the conventional MCS, we modify the momentum conservation principle to be more general, accommodating a non-negative wet cross-sectional area due to irregular geometry. We successfully conduct numerous numerical simulations by examining various scenarios involving one-layer and two-layer flow through irregularly shaped channels or structures. Our results show that the correct surface wave profile generated by a one-dimensional dam break through the triangular obstacle in the open channel can be simulated very well. Comparison with the existing experimental data seems promising although some disparities are being found due to dispersive phenomena with RMSE less than 5%. Furthermore, our scheme is successfully extended to simulate the steady sub-maximal exchange in two-layer flows using specific boundary conditions. The alignment between the submaximal numerical results with exchange flow theory is noticeable in the interface profile, characteristics of flow conditions and the flux values achieved when the steady situation occurs. These satisfying results indicate that our proposed numerical model can be used for practical needs involving various flow situations both one and two-layer cases

Residual stresses for a new class of transversely isotropic nonlinear elastic solid

We study the response of a class of transversely elastic bodies, wherein the Green–Saint Venant strain tensor is a function of the second Piola–Kirchhoff stress tensor, when the body is residually stressed. The notion of such non-Cauchy elastic bodies being transversely isotropic is defined in Rajagopal (Mech. Res. Commun. 64, 2015, 38–41), and by a body being residually stressed, we mean the interior of the body is not in a stress-free state although the boundary is free of traction as considered by Coleman and Noll (Arch. Ration. Mech. Anal. 15, 1964, 87–111) and by Hoger (Arch. Ration. Mech. Anal. 88, 1985, 271–289).

Buckling analysis of thin-walled laminated composite or functionally graded sandwich I-beams using a refined beam theory

This study presents the buckling and lateral torsional buckling behaviors of thin-walled laminated composite or functionally graded sandwich I-beams. A refined beam model (RBT) based on the 3D Saint-Venant (SV) solution is used in this study to formulate the problem. This model allows for a realistic analysis of beams with arbitrary cross-sections and is free from the limitations of classical beam theories. The displacement models establish a consistent 1D beam theory that accurately captures the essence of the cross-section’s nature. The governing models consider cross-section deformations, including in and out of plane warpings and distortions derived from the 3D SV solution associated with the cross-section vibrational behavior. Furthermore, a user-friendly numerical tool called CSB (cross-section and beam analysis) is used to facilitate the implementation of this method. The numerical results are examined in detail and compared with previous works to investigate the influence of boundary conditions, angle-ply, shear effects, span-to-height ratio, and material distribution on the critical buckling loads of thin-walled I-beams. The results demonstrate that the RBT models accurately and efficiently analyze thin-walled I-beams under different loads and boundary conditions. Furthermore, some of the new results are presented as reference values for the future.

Comprehensive mathematical model for efficient and robust control of irrigation canals

Linear control-oriented models are important to represent canal dynamics for designing controllers. This study focuses on hydraulic control structure (gate) modelling to address the complex interdependent behavior inherent in irrigation canals. A comprehensive mathematical model that incorporates the water level with gate-opening to model discharge is introduced for single and multiple canal pool scenarios. The proposed model captures the hydraulic coupling within and among canal pools, a key finding. The model is evaluated extensively under uniform and non-uniform flows across three distinct canals, highlighting the model's applicability to various systems. The uncertainty inherent within the nominal model is also assessed for varying operating conditions and hydraulic parameters. The proposed model is compared with the existing and the Saint-Venant (SV) model, showing improved accuracy in water-level predictions. This advancement in hydraulic modelling contributes to adaptable canal models essential in developing robust controllers to enhance water management in irrigation canals.

Damage prediction and improvement method based on cutting mode of circular empty hole

Based on the theory of empty hole effect of cutting blasting, the Hopkinson effect and Saint–Venant principle are integrated to establish a two-dimensional calculation model of dynamic stress evolution of the holes wall, and then the dynamic fracture mechanism and damage distribution mode of the rock mass in the cutting area under the action of longitudinal waves are predicted. The results of the calculation and numerical simulation are verified by experiments, and the results show that: The time-varying stress function of the circular cavity wall conforms to the periodic dynamic evolution of the trigonometric function, and the theoretical calculation is consistent with the simulation results. Through the calculation of the round holes cut model and the square empty hole cut model, the change of the shape of the holes in the cut area changes the failure form of the surrounding rock mass. The circular empty hole wall is affected by the stress wave to produce "interval ring" destruction, and the effect of the reflected stretch wave is inhibited. The large range of rock mass in the square empty hole wall produces tensile and shear failure, and the rock mass collapses inward under the influence of the second stage stress. Among them, the empty space utilization rate of the square empty hole model is about 8.5 times that of the circular holes model. Vibration monitoring in the center of the cutting area shows that the vibration effect of the circular empty hole is larger than that of the square empty hole, and the proportion of rock breaking energy is lower.

Galerkin-Kantorovich variational method for solving saint venant torsion problems of rectangular bars

The unrestrained torsional analysis of bars is an important theme in elasticity theory, first solved by Saint-Venant using semi-inverse methods. It has been considered and solved by several others using analytical methods and numerical procedures due to the importance in the design of machine parts under torsional moments. In this paper, the Saint Venant torsion problem is solved for rectangular prismatic bars using Galerkin-Kantorovich variational method (GKVM). The work presents a detailed theoretical framework of the problem, deriving using first principles considerations the stress compatibility equation in terms of the Prandtl stress function ϕ(x,y). The derived domain equation which is required to be satisfied over the rectangular cross-sectional domain is a partial differential equation of the Poisson type. GKVM is adopted as the solution method for finding the solution to the domain equation. The unknown Prandtl stress function ϕ(x,y) is assumed, following Kantorovich method to be a product of an unknown function for fx sought to minimize the Galerkin-Kantorovich variational functional (integral) (GKVF) and a known function (y2-b2) which satisfies the boundary conditions at all boundary points in the y-direction, that is, at y=±b. The resulting GKVF is a simplified functional whose integral is a second order inhomogeneous ordinary differential equation (ODE) in fx. The integrand is solved to find fx leading in a full determination of the Prandtl stress function. The expression for stresses, torsional moments and torsional parameters are then found and they satisfy the boundary conditions and the domain equation. The results for the torsional moments and torsional parameters are identical to previous results obtained using double finite sine transform method (DFSTM), and analytical methods. The merit of GKVM is that it has led to the exact solution of the unrestrained torsion problems.

Solving Saint Venant torsion problems for rectangular beams using single finite Fourier sine transform method

This research presents the single Fourier sine transform method (SFSTM) for solving the Saint Venant torsion problem of rectangular prismatic bars. The problem is a common theme in the theory of elasticity of unrestrained torsion which was previously expressed by Prandtl using Prandtl stress functions ϕ(x,y) as a Poisson type nonhomogeneous partial differential equation (PDE) called the stress compatibility equation. In this work the SFSTM was applied to the stress compatibility equation, converting the PDE to an easier to solve ordinary differential equation (ODE) in the transformed domain. The boundary conditions were used to find the integration constant and inversion was used to find the solution in the physical domain. The non vanishing stresses and torsional moments were thus found as a single series of infinite terms with rapid convergence. The maximum stresses and moments were found in standard form in terms of torsional parameters which were tabulated for various ratios of the cross-sectional dimensions. A comparison of the torsional parameters with previous results show that the present results are identical with previous results illustrating the accuracy of the SFSTM used. The sine kernel of the SFSTM satisfies the boundary conditions of the problem and contributed to the exact solution obtained. The SFSTM simplified the PDE to an ODE which is simpler to solve.

Preissmann Four-Point Methods for Solution of Simplified Saint-Venant Equations Applied to Flood Routing in Prismatic Open Channels

This research goal to compare the flow properties in the rectangular and trapezoidal open channels by examining the influence of the channel side slope is depicted by simplified Saint Venant Equations. The solution of these equations has been completed numerically by using Preissmann four-point scheme. The model is simulated using the Matlab application to point out the flow properties. The proposed model is validated by the model without simplification which was selected from the literature. The validation outcomes indicate that in common, the simulation outcomes of the two models have a good agreement. The simulation results show that the greater the slope of the channel side, the greater the peak discharge and the greater the time shift. The analysis emphasizes how channel geometry influences flow behavior, indicating that trapezoidal channels, with inclined side slopes z, yield slightly higher peak discharges compared to rectangular ones. For z = 0, discharge of peak Q = 7.38 m3/s and t = 18 s. For z = 2, discharge of peak Q = 7.39 m3/s and t = 21 s. For z = 4, discharge of peak Q = 7.45 m3/s and t = 23 s. For z = 6, discharge of peak Q = 7.51 m3/s and t = 24 s.

Tsunami Inundation Modelling in a Built-In Coastal Environment with Adaptive Mesh Refinement: The Onagawa Benchmark Test

In tsunami studies, understanding the intricate dynamics in the swash area, characterised by the shoaling effect, remains a challenge. In this study, we employed the adaptive mesh refinement (AMR) method to model tsunami inundation and propagation in the Onagawa town physical flume experiment. Using the open-source flow solver Basilisk, we implemented the Saint-Venant (SV) equations, Serre–Green–Naghdi (SGN) equations, and a nonhydrostatic multilayer (ML) extension of the SGN equations. A hydraulic bore tsunami-like wave was used as the input boundary condition. The objective was to assess the efficiency of the AMR method with nonhydrostatic tsunami models in overcoming limitations in 2D and quasi-3D models in flume experiments, particularly with respect to improving accuracy in arrival time and run-up detection. The results indicate improved performance of the SGN and SV models in determining tsunami arrival times. The ML model demonstrated enhanced wave run-up simulations on complex built-in terrain. The refined roughness coefficient determined using the ML solver captured the arrival time well in the northern section of the Onagawa model, albeit with a 1 s delay. The AMR method offered a computationally stable solution with an 86.3% reduction in computational time compared to a constant grid. While effective, the nonhydrostatic models entail the use of a great deal of computational resources.

A 3D distributed plasticity beam–column element for metal structures considering tangential stresses and warping with minimal DOFs

The fiber model evaluates the normal stress at a number of points over the section for a given strain increment following the plane section assumption and, by integration, axial force and bending moments. The interaction with tangential stresses is usually neglected at the point level due to the inaccurate tangential strains of the kinematics. This work proposes a generalization of the fiber model able to capture automatically the interaction among all stress components. A preliminary cross-section analysis based on the Saint Venant problem provides an accurate 3D strain as a function of the section generalized strains. This field, accurate also in the inelastic case, is exploited to impose at each section point a 3D von Mises elasto-plastic law, obtaining by integration all the resultants and moments with a full interaction. Non-uniform warping is also easily included. The section model is implemented in a mixed 3D beam–column finite element with equilibrated stress field, accurate with a minimal mesh. Numerical tests show the excellent prediction of the proposal compared to analytical and solid FEM solutions also for structures not flexure-dominated. Its efficiency, on the same order as a standard fiber model, makes the approach suitable also for large buildings.

A hybrid dislocation deforming two welded anisotropic crystals

ABSTRACT The elastic field of an angular dislocation line that has a branch placed in a crystal of a bi-material while the other branch is along a strongly bound or welded interface, is found when the two crystals have anisotropic properties. Since the field of this angular dislocation depends on the elastic constants of the two crystals, it can be called hybrid dislocation (HD). To face this problem already treated using isotropic elasticity, the analysis first formulates the elastic field of a closed dislocation loop whose surface S is that of a circular sector centred in a point O of the interface. By virtue of the Saint Venant principle, when the circular segment moves far away from O, the field induced by the loop near O tends towards that of a HD. The calculation is possible thanks to the knowledge of the Green’s tensor of the bi-material which has already been the subject of several works. Thanks to the use of polar coordinates, the solution is reduced to convenient line-integral expressions which make the calculation faster. This work complements a previous approach to the HD problem based on the integral formalism. By combining two HD’s, it also gives the solution to the problem of an interfacial angular dislocation. Some applications illustrate anisotropy effects and highlight an inability of integral formalism to deal with a interfacial branch strictly in the interface.

F28: A novel coupling strategy for 1D/2D hydraulic models for flood risk assessment of the Mekong Delta

Abstract Coupling models of different dimensions is one of the most important yet under-represented challenges. This paper introduces a new modeling strategy to streamline a more flexible and effective integrated one-dimensional (1D)/two-dimensional (2D) model for floodplains along lowland rivers. The 1D model, utilizing the finite volume method, solves the Saint–Venant equations, while the 2D mesh employs unstructured quadrilateral elements. The two strategies couple the 1D/2D models: direct 1D/2D connection by the law of mass conservation at supernode, and lateral 1D/2D model connection by spillways at riverbank. The coupling strategy in F28 guarantees the water balance and the conservation of momentum at the integrated 1D/2D nodes. The model was applied to the Mekong Delta to address the capacity of hydrodynamic simulations integrating various water infrastructures. Results showed that the developed model has a strong potential to be applied to other lowland rivers worldwide with complex infrastructures.

An advanced check‐dam sedimentation module: Laboratory validation and implementation in a distributed sediment yield model for field application

AbstractCheck‐dams are structures used extensively around the world for soil and water conservation. However, existing models for check‐dams are unable to simulate the Sediment Trap Efficiency (STE) at the catchment scale. A numerical model was developed to simulate the SEdiment Deposition upstream of Check‐Dams (SEDCD) and integrated into a distributed sediment yield model, the Digital Yellow River Model (DYRIM). Two versions of the SEDCD model were evaluated: the SV version which used the Saint–Venant equation to simulate the unsteady flow, and the BW version, which used a modified backwater equation (based on a quasi‐steady approximation) to improve computational efficiency. Sediment deposition and the associated bed profile adjustment were simulated with the sediment conservation equation and the non‐equilibrium suspended sediment transport equation. The SEDCD model was first validated in the laboratory using experimental data from a scale‐down check‐dam. The bed profiles predicted using both versions of the SEDCD model showed good agreement with the observations, with NSE values over 0.9 in most profiles. When integrated into the DYRIM and applied to the Xiaoli River Basin (818 km2) on the Loess Plateau, which has 183 active check‐dams, the SEDCD‐DYRIM combination predicted the STE for an extreme rainstorm event in 2017 with good accuracy and high computational efficiency. The SEDCD‐BW‐DYRIM simulated the hourly discharge and sediment concentration with high accuracy (NSE values of 0.79 and 0.71, respectively) and provided single‐event STEs (R2 value of 0.99) comparable to those of the SEDCD‐SV model, with an approximately 30 times faster runtime efficiency than the SEDCD‐SV model. The SEDCD‐BW model is a powerful and efficient tool to assess the effect of check dams on sediment dynamics at the catchment scale.

Sonics: develop intuition on biomechanical systems through interactive error controlled simulations

We describe the SOniCS (SOFA + FEniCS) plugin to help develop an intuitive understanding of complex biomechanics systems. This new approach allows the user to experiment with model choices easily and quickly without requiring in-depth expertise. Constitutive models can be modified by one line of code only. This ease in building new models makes SOniCS ideal to develop surrogate, reduced order models and to train machine-learning algorithms for enabling real-time patient-specific simulations. SOniCS is thus not only a tool that facilitates the development of surgical training simulations but also, and perhaps more importantly, paves the way to increase the intuition of users or otherwise non-intuitive behaviors of (bio)mechanical systems. The plugin uses new developments of the FEniCSx project enabling automatic generation with FFCx of finite-element tensors, such as the local residual vector and Jacobian matrix. We verify our approach with numerical simulations, such as manufactured solutions, cantilever beams, and benchmarks provided by FEBio. We reach machine precision accuracy and demonstrate the use of the plugin for a real-time haptic simulation involving a surgical tool controlled by the user in contact with a hyperelastic liver. We include complete examples showing the use of our plugin for simulations involving Saint Venant–Kirchhoff, Neo-Hookean, Mooney–Rivlin, and Holzapfel Ogden anisotropic models as supplementary material.

Large deformation of hyperelastic modified Timoshenko–Ehrenfest beams under different types of loads

In this paper, a nonlinear finite element formulation for a hyperelastic, modified Timoshenko–Ehrenfest beam with geometrical and material nonlinearities is developed for the first time. A new five-parameter beam element is introduced. The parameters contain displacement, values of difference vector and a through-the-thickness scalar value. Moreover, a new procedure is employed to apply the moment to the beam. The constitutive formulation is derived for Saint Venant–Kirchhoff (SVK) as well as the compressible neo-Hookean (n-H) hyperelastic model. The beam is subjected to different loads such as dead load, point and distributed loads, and follower pressure. To demonstrate the applicability of the formulations, several examples are solved. The results reveal that this formulation can capture the previous results reported in the literature. Furthermore, a comparative study is done between two hyperelastic models. It is demonstrated that in the case of large rotation of the beam, both Saint Venant–Kirchhoff and neo-Hookean models show the same behavior. However, in case of large deformation with large strains of the beam, the Saint Venant–Kirchhoff model behaves stiffer than the neo-Hookean hyperelastic model.

Modeling unsteady and steady 1D hydrodynamics under different hydraulic conceptualizations: Model/Software development and case studies

The stage-discharge relationship is affected by hysteresis, especially in 1D river cross-sections with very mild slopes, and/or where backwater effects occur. To model these cases, hydraulic property (HP) estimations (e.g., hydraulic radius and conveyance) are usually required as closed-form functions of stage. However, these are typically unavailable for complex cross-sections with overbanks. For no sediment-laden flows, we created the HydroHP - 1D tool to solve the 1D Saint Venant Equations aided by the cross-section HP stored in tables. It also simulates irregular and composite hydraulic cross-sections, allows hydrodynamic simulation using the single-channel method (SCM), and proposes the divided channel method (DCM) for steady and unsteady flow analysis. A wide range of applications in rivers such as the Little Washita River and the Upper Negro River, and comparisons with NOAA normal depth solver, HEC-RAS, and with an analytical solution shows a promising scenario of applying HydroHP - 1D to estimate 1D steady and unsteady hydrodynamics.