Small Assumption Research Articles

An electro-elastic contact problem with an internal state variable

We consider a mathematical model which describes the quasistatic process of contact between a piezoelectric body and an electrically conductive foundation. General models for elastic materials with piezoelectirec effect, called electro-elastic materials. This paper is devoted to the study of the model involving a frictionless contact between an electro-elastic body characterized by long memory and a conductive adhesive foundation. The process is mechanically quasistatic and electrically static. We model the material with a general nonlinear electro-elastic constitutive law with internal state variable and the contact with normal compliance and unilateral constraint, where the adhesion is taken into account and a regularized electrical conductivity condition. The main feature of these models consists of the fact that their variational formulation is given by a history-dependent quasivariational inequality for the displacement field and a linear variational equation for the electric potential. A weak formulation for the model is given in the form of a coupled system for the displacement, the stress, the electric displacement, the electric potential, the bonding and the variable state internal fields. We derive a variational formulation for the problem and then, under a smallness assumption on the data, we prove the existence of a unique weak solution to the model. We also investigate the behavior of the solution with respect the electric data on the contact surface and prove its unique weak solution. The proof is based on nonlinear evolution equations with monotone operators, differential equations and Banach fixed point arguments.

Well‐posedness and large‐time behavior for the two‐phase system with magnetic field in critical Besov spaces

AbstractThis paper is dedicated to the study of the two‐phase system with magnetic field in multi‐dimensional spaces . We investigate the local and global well‐posedness of strong solutions to the Cauchy problem in critical Besov spaces. Moreover, a Lyapunov‐type energy argument can be developed, which leads to the time‐decay estimates of solutions without additional smallness assumptions. We improve the previous effects obtained by Wen and Zhu (JDE‐2018), Wang and Cheng (AMS‐2022), where they established the global well‐posedness of strong solutions in the ‐norm Sobolev spaces.

Nitsche method for Navier–Stokes equations with slip boundary conditions: convergence analysis and VMS-LES stabilization

In this paper, we analyze Nitsche’s method for the stationary Navier–Stokes equations on Lipschitz domains under minimal regularity assumptions. Our analysis provides a robust formulation for implementing slip (i.e., Navier) boundary conditions in arbitrarily complex boundaries. The well-posedness of the discrete problem is established using the Banach Nečas–Babuška and Banach fixed point theorems under standard small data assumptions. We also provide optimal convergence rates for the approximation error. Furthermore, we propose a quasi-static VMS-LES formulation with Nitsche for the Navier–Stokes equations with slip boundary conditions to address the simulation of incompressible fluids at high Reynolds numbers. We validate our theory through several numerical tests in well-established benchmark problems.

Using fractal voids to understand Mode I compressive fracture in brittle materials – A two-dimensional analysis

Unlike cases of direct or indirect tensile loading, analysis of Mode I fracture in cases of compressive loading must necessarily consider two-dimensional flaw geometries. The consequent analytical complications have resulted in little theoretical evolution towards understanding Mode I fracture in compressive loading. Researchers have recently observed that the linear elastic (LE) stress fields surrounding two-dimensional flaws in compression appear to have indicative value regarding Mode I fracture, but no rational framework has yet been proposed for their interpretation. Herein it is demonstrated that by idealizing void surfaces as fractal, and using what will be called the “small flaw assumption,” LE stress fields can be interpreted to obtain significant insight regarding brittle Mode I compressive fracture. Using fractal voids, a rational and consistent explanation for the satisfaction of the stress and energy criteria at the surface of two-dimensional flaws is presented. The discussion resolves many typical theoretical issues in the literature in a manner consistent with fundamental Griffith theory, and accounts for size and shape effects. The framework further enables the computationally efficient prediction of peak KI values associated with the propagation of line cracks from two-dimensional flaws from a brittle medium subject to macroscopic uniaxial compression via LE stress fields.

Analytical Formulation and Optimization of the Initial Morphology of Double-Layer Cable Truss Flexible Photovoltaic Supports

With the rapid development of the photovoltaic industry, flexible photovoltaic supports are increasingly widely used. Parameters such as the deflection, span, and cross-sectional dimensions of cables are important factors affecting their mechanical and economic performance. Therefore, in order to reduce steel consumption and cost and improve application value, it is crucial to design and optimize their initial morphology. In this paper, the mechanical behavior of a single-cable structure is introduced, and the simplified analytical formulations for internal force and displacement are deduced based on the geometric nonlinear characteristics and small strain assumption of the flexible photovoltaic supports. On this basis, the analytical expressions for the cable force and displacement of a convex prestressed double-layer cable truss flexible photovoltaic support structure under a uniform load are derived, and the correctness of the analytical formulations is verified by comparing the values with the finite element analysis results. In order to reduce the construction costs of the flexible photovoltaic support, a mathematical model for optimizing the initial structure’s morphology is established according to the analytical formulations. The initial morphology of the double-layer cable truss flexible photovoltaic support is optimized, and the optimization results of different deflection deformation limits and whether the lower load-bearing cable is allowed to relax are compared. The results indicate that the errors of the displacement formulation and cable force formulation, when compared with the finite element results, are less than 3% and 4%, respectively, which verifies the accuracy of the analytical formulations. By analyzing the cable force and displacement of the structure under static action, it is suggested that the deflection limit of the double-layer cable truss structure should be 1/100 of the single span. The lower load-bearing cables of the double-layer cable truss flexible photovoltaic support are highly susceptible to relaxation under wind suction loads, and, by comparing the optimization results, it is suggested that slack should be allowed in the lower load-bearing cables for a better economic effect. When choosing the most economical structure morphology, it is recommended that the total height of the mid-span struts should be 1/20~1/15 of the single span. The analytical formulation and the mathematical model for the optimization of the initial morphology proposed in this paper can provide certain theoretical references and bases for the design of practical engineering projects and play an important role in promoting its application and promotion.

Cost Optimisation of Individual-Based Institutional Reward Incentives for Promoting Cooperation in Finite Populations

In this paper, we study the problem of cost optimisation of individual-based institutional incentives (reward, punishment, and hybrid) for guaranteeing a certain minimal level of cooperative behaviour in a well-mixed, finite population. In this scheme, the individuals in the population interact via cooperation dilemmas (Donation Game or Public Goods Game) in which institutional reward is carried out only if cooperation is not abundant enough (i.e., the number of cooperators is below a threshold 1≤t≤N-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le t\\le N-1$$\\end{document}, where N is the population size); and similarly, institutional punishment is carried out only when defection is too abundant. We study analytically the cases t=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t=1$$\\end{document} for the reward incentive under the small mutation limit assumption and two different initial states, showing that the cost function is always non-decreasing. We derive the neutral drift and strong selection limits when the intensity of selection tends to zero and infinity, respectively. We numerically investigate the problem for other values of t and for population dynamics with arbitrary mutation rates.

Convergence analysis of an efficient scheme for the steady state second grade fluid model

We are interested in studying the stationary second grade fluid model in a bounded domain in R2. To approximate the solution of the continuous model, we propose a fully decoupled numerical scheme based on a splitting method combined with the use of the Grad–Div operator. This approach allows the complete decoupling of the three variables: velocity, pressure and vorticity. Each variable is computed using an iterative procedure, with the pressure step involving a simple L2-projection.We provide a proof of the convergence of the scheme to the continuous problem under smallness assumptions on the data. This theoretical analysis ensures the reliability of our method in approximating the behavior of the stationary second grade fluid model.Finally, we present several numerical tests to validate our approach. These tests illustrate the effectiveness and efficiency of our scheme in various scenarios, highlighting its potential applicability to a wide range of problems involving second grade fluids.

Analysis of a dynamic contact problem with long memory

We examine a mathematical framework detailing a dynamic frictional contact involving thermo-viscoelastic materials characterized by long memory and damage. The contact is modeled by the normal compliance condition with Coulomb's friction law, while surface wear is also factored in. The model was in the form of a coupled system which includes an evolution equation written in inclusion form for the frictional contact, parabolic variational inequality for the damage field, nonlinear differential equation for the temperature field and an equation for the wear particles and an evolution. A variational formulation for the model was derived and the existence of the unique weak solution established, under smallness assumptions on a part of the problem data. The proof used various results from the theory of evolution inequalities and repeated fixed-point arguments.

Localization of Beltrami fields: Global smooth solutions and vortex reconnection for the Navier-Stokes equations

We introduce a class of divergence-free vector fields on R3 obtained after a suitable localization of Beltrami fields. First, we use them as initial data to construct unique global smooth solutions of the three dimensional Navier-Stokes equations. The relevant fact here is that these initial data can be chosen to be large in any critical space for the Navier–Stokes problem, however they satisfy the nonlinear smallness assumption introduced in [10]. As a further application of the method, we use these vector fields to provide analytical example of vortex-reconnection for the three-dimensional Navier-Stokes equations on R3. To do so, we exploit the ideas developed in [13] but differently from this latter we cannot rely on the non-trivial homotopy of the three-dimensional torus. To overcome this obstacle we use a different topological invariant, i.e. the number of hyperbolic zeros of the vorticity field.

Acceleration of a wave-structure interaction solver by the Parareal method

Potential flow theory-based solvers are commonly used in ocean engineering to investigate the interactions between ocean waves and floating bodies. Depending on assumptions, several methods have been proposed. Among them, the Weak-Scatterer method is an interesting trade-off in the sense that this approach is not limited in theory by the small wave amplitudes and small body motions assumptions of linear methods. Moreover, this approach is in practice more stable than the fully non-linear methods. An implementation of the Weak-Scatterer method is the WS-CN code (Letournel, 2015; Chauvigné, 2016; Wuillaume, 2019).The computational time of the WS-CN code which is considered in the present study is relatively long for engineering purposes. In order to reduce it, the present paper presents an implementation of the Parareal method in the WS-CN code. The Parareal method is an algorithm for parallelizing a simulation in time that can accelerate the complete simulation (Lions, 2001) . This is a key difference in comparison to other acceleration techniques which have been studied in the literature (e.g. the Fast Multipole Method (FMM), the precorrected Fast Fourier Transform (pFFT) method, … ). To the authors’ knowledge, the present study is the first to couple the Parareal method to a potential flow theory-based wave-structure interaction solver. It is shown that the method can significantly reduce the computational time for small wave steepness, but that the performance decreases rapidly with increasing steepness.

On Error Estimates of a discontinuous Galerkin Method of the Boussinesq System of Equations

Abstract In this paper, we propose and analyze a discontinuous Galerkin finite element method for solving the transient Boussinesq incompressible heat conducting fluid flow equations. This method utilizes an upwind approach to handle the nonlinear convective terms effectively. We discuss new a priori bounds for the semidiscrete discontinuous Galerkin approximations. Furthermore, we establish optimal a priori error estimates for the semidiscrete discontinuous Galerkin velocity approximation in L 2 \mathbf{L}^{2} and energy norms, the temperature approximation in L 2 L^{2} and energy norms and pressure approximation in L 2 L^{2} -norm for t > 0 t>0 . Additionally, under the smallness assumption on the data, we prove uniform in time error estimates. We also consider a backward Euler scheme for full discretization and derive fully discrete error estimates. Finally, we provide numerical examples to support the theoretical conclusions.

Topology optimization of curved thick shells using level set method and non-conforming multi-patch isogeometric analysis

We present a novel framework for topological shape optimization of curved non-conforming multi-patch and trimmed thick-shells subjected to external loads. Our method integrates the level set method (LSM) with a diffuse interface, a Hadamard shape derivative, and multi-patch isogeometric analysis (IGA) into a gradient descent algorithm to systematically capture the evolution of the shape. This integration enables us to directly manipulate CAD-compatible geometries and analysis techniques and to obtain the results as a CAD surface. The novelty lies in the utilization of multi-patch IGA models based on NURBS functions, which allows us to simultaneously maximize the stiffness and minimize the volume of the shell by searching for an optimal material distribution within its middle surface. The material is modeled under a small strain assumption in linear elasticity using a Reissner–Mindlin kinematic shell model in plane stress. The effectiveness of our approach is demonstrated on several curved conforming and non-conforming multi-patch geometries in 3D.

How to fight fake papers: a review on important information sources and steps towards solution of the problem.

Scientific fake papers, containing manipulated or completely fabricated data, are a problem that has reached dramatic dimensions. Companies known as paper mills (or more bluntly as "criminal science publishing gangs") produce and sell such fake papers on a large scale. The main drivers of the fake paper flood arethe pressure in academic systems and (monetary) incentives to publish in respected scientific journals and sometimes the personal desire for increased "prestige." Published fake papers cause substantial scientific, economic, and social damage. There are numerous information sources that deal with this topic from different points of view. This review aims to provide an overview of these information sources until June 2024. Much more original research with larger datasets is needed, for example on the extent and impact of the fake paper problem and especially on how to detect them, as many findings are based more on small datasets, anecdotal evidence, and assumptions. A long-term solution would be to overcome the mantra of publication metrics for evaluating scientists in academia.

An exactly divergence-free hybridized discontinuous Galerkin method for the generalized Boussinesq equations with singular heat source

The purpose of this work is to propose and analyze a hybridized discontinuous Galerkin (HDG) method for the generalized Boussinesq equations with singular heat source. We use polynomials of order k, k − 1 and k to approximate the velocity, the pressure and the temperature. By introducing Lagrange multipliers for the pressure, the approximate velocity field obtained by the HDG method is shown to be exactly divergence-free and H(div)-conforming. Under a smallness assumption on the problem data and solutions, we prove by Brouwer’s fixed point theorem that the discrete system has a solution in two dimensions. If the viscosity and thermal conductivity are further assumed to be positive constants, a priori error estimates with convergence rate O(h) and efficient and reliable a posteriori error estimates are derived. Finally numerical examples illustrate the theoretical analysis and show the performance of the obtained a posteriori error estimator.

Hypersonic similarity for steady compressible full Euler flows over two-dimensional Lipschitz wedges

We establish the optimal convergence rate to the hypersonic similarity law, which is also called the Mach number independence principle, for steady compressible full Euler flows over two-dimensional slender Lipschitz wedges. Mathematically, it can be formulated as the comparison of the entropy solutions in BV∩L1 between the two initial-boundary value problems for the compressible full Euler equations with parameter τ>0 and the hypersonic small-disturbance equations (the scaled compressible full Euler equations with parameter τ=0) with curved characteristic boundaries. We establish the L1–convergence estimate of these two solutions with the optimal convergence rate, which justifies the Van Dyke's similarity theory rigorously for the compressible full Euler flows. This is the first mathematical result on the comparison of two solutions of the compressible Euler equations with characteristic boundary conditions. To achieve this, we first employ the special structures of the two systems and establish the global existence and the L1–stability of the entropy solutions via the wave-front tracking scheme under the smallness assumption on the total variation of both the initial data and the tangential slope function of the wedge boundary. Based on the L1–stability properties of the approximate solutions to the scaled equations with parameter τ>0, a uniform Lipschtiz continuous map with respect to the initial data and the wedge boundary is obtained, which is the first time for the characteristic boundary conditions. Next, we compare the solutions given by the Riemann solvers of the two systems by taking the boundary perturbations into account case by case. Then, for a given fixed hypersonic similarity parameter, as the Mach number tends to infinity, by employing the Lipschitz continuous properties of the map, we establish the desired L1–convergence estimate with the optimal convergence rate. Finally, we show the optimality of the convergence rate by investigating a special solution.

The Symmetric Dual Inductor Hybrid Converter for Direct 48V-to-PoL Conversion

This work introduces the symmetric dual inductor hybrid (SDIH) dc-dc converter topology, which is suitable for large conversion ratios where regulation is required, such as direct 48 V to point-of-load (PoL) applications. A dickson-type switched capacitor network is used to effectively produce two interleaved PWM outputs with a greatly reduced voltage amplitude relative to the input voltage, allowing the subsequent magnetic volume to be reduced while retaining modest switching frequencies. Distinct from related variations, part count is significantly reduced while both even and odd order switched capacitor networks can be used with straightforward split-phase control; allowing either network type to achieve complete soft-charging of all flying capacitors. Additionally, charge flow is uniformly distributed through all elements, with equal capacitor and inductor values being preferred. Subsequently this topology is expected to simplify component selection, improve electrical and thermal performance, and reduce cost. Furthermore, analysis is presented that calculates precise phase durations without making small ripple assumptions, revealing up to a 75% timing error in cases where either inductor or capacitor ripple is ignored. Finally, a discrete prototype validates this analysis and demonstrates very high measured power densities of 1,029 W/in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{3}$</tex-math></inline-formula> , 754 W/in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{3}$</tex-math></inline-formula> , and 663 W/in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{3}$</tex-math></inline-formula> for 48 V input and regulated output voltages of 3 V, 2 V, and 1 V, respectively, while switching at a frequency of 750 kHz.

Global well‐posedness of the 3D hydrostatic magnetohydrodynamics equations

In this paper, the three‐dimensional hydrostatic magnetohydrodynamic (HMHD) equations are considered on a thin domain. We showed the global existence and uniqueness (regularity) of strong solutions to the three‐dimensional incompressible HMHD equations without any small assumption on the initial data. More precisely, there exists a unique strong solution globally in time for any given ‐smooth initial data.

A multiscale micromechanical progressive elastic-damage model for cementitious composites featuring superabsorbent polymer (SAP)

A multiscale micromechanics-based progressive damage model is developed to investigate the overall mechanical behavior and the interfacial microcrack evolutions of the cementitious composites featuring superabsorbent polymer (SAP) under uniaxial tension. Elastic properties, progressive damage process, and homogenization procedure of cementitious composites are systematically integrated in this model. The effective elastic moduli of the composites are determined based on a multiscale micromechanical framework. According to the small strain assumption, the total strain tensor and the elastic-damage compliance tensor are additively decomposed into elastic and damage-induced components. The damage-induced strains and compliances are then deduced from micromechanics. To characterize the progressive elastic-damage induced by microcracks, stages of microcrack propagation are identified from the interface contact stress and the matrix cleavage stress. The complex potentials and stress intensity factors for kinked interface cracks are derived from the distributed dislocations method. By implementing the homogenization process, the macroscopic mechanical behavior is obtained from the micro/mesoscale. The results indicate that the material parameters have clear mechanical significance. Different parameters, such as the SAP addition ratio, aggregate content, initial interfacial crack size, and initial interfacial crack location, are revealed to be influential in the overall mechanical behavior of the composites. The proposed model can be generalized to other particle-reinforced composites with different constituent properties, which can potentially contribute to the design and optimization of durable composites.

Data-based modelling of arrays of wave energy systems: Experimental tests, models, and validation

One of the key steps towards economic feasibility of wave energy conversion technology concerns scaling up to farms of multiple devices, in the attempt to reduce installation costs by sharing infrastructure, and a consequent drop in levelised cost of energy. Moreover, whenever wave energy systems are deployed in proximity (in so-called arrays), the exploitation of the hydrodynamic interactions between single devices is fully enabled, potentially increasing the final energy outcome. To achieve this in real (operational) time, the employed energy-maximising control strategies require control-oriented array models, able to efficiently describe the dynamics of these interconnected systems in a representative fashion. This can be, nonetheless, a difficult task when considering first principles alone, under small motion assumptions, for modelling purposes. Recognising the uncertainty associated to array numerical models obtained from the linearisation of simplified system equations around their equilibria, this paper presents models of several array configurations identified following a frequency domain approach on the basis of experimental data. Tailored tests on laboratory-scale devices have been designed and conducted in the Aalborg University (Denmark) wave tank facility, with the purpose of performing representative system identification of the wave energy systems arrays. The obtained models are validated on different representative sea states configurations, in controlled and uncontrolled motion operational conditions. The validation results are fully discussed and analysed in terms of standard error measures and time lag, while the obtained models are made freely accessible via a linked repository (named OCEAN), in the attempt to openly provide validated models for different array configurations.

On the stability of stationary compressible Navier–Stokes flows in 3D

AbstractThis paper studies the stability of the stationary solution of the compressible Navier–Stokes equation in the 3D whole space with an external force which decays at spatial infinity. We obtain the global existence result of the non-stationary problem under the smallness assumptions on the initial perturbation around the small stationary solution. We also derive the time decay rates of the perturbations under the smallness assumption on the initial perturbations, and show the optimality of the time decay rates. The proofs are based on the combination of the spectral analysis and energy method in the framework of Besov spaces. The time-space integral estimate for the linearized semigroup around the constant state in some Besov spaces plays a crucial role in the proofs.