Existence of Monotone Positive Solution for a Fourth-Order Three-Point BVP with Sign-Changing Green’s Function

This paper is concerned with the following fourth-order three-point boundary value problem , where , we discuss the existence of positive solutions to the above problem by applying to the fixed point theory in cones and iterative technique.

Multi-Objective Optimization of Aluminum Alloy Electric Bus Frame Connectors for Enhanced Durability

The widespread adoption of aluminumalloy electric buses,known for their energy efficiency and eco-friendliness,faces a challenge due to the aluminum frame’s susceptibility to deformation compared to steel.This issue is further exacerbated by the stringent requirements imposed by the flammability and explosiveness of batteries,necessitating robust frame protection.Our study aims to optimize the connectors of aluminum alloy bus frames,emphasizing durability,energy efficiency,and safety.This research delves into Multi-Objective Coordinated Optimization(MCO)techniques for lightweight design in aluminum alloy bus body connectors.Our goal is to enhance lightweighting,reinforce energy absorption,and improve deformation resistance in connector components.Three typical aluminum alloy connectors were selected and a design optimization platform was built for their MCO using a variety of software and methods.Firstly,through three-point bending experiments and finite element analysis on three types of connector components,we identified optimized design parameters based on deformation patterns.Then,employing Optimal Latin hypercube design(OLHD),parametric modeling,and neural network approximation,we developed high-precision approximate models for the design parameters of each connector component,targeting energy absorption,mass,and logarithmic strain.Lastly,utilizing the Archive-based Micro Genetic Algorithm(AMGA),Multi-Objective Particle Swarm Optimization(MOPSO),and Non-dominated SortingGenetic Algorithm(NSGA2),we explored optimized design solutions for these joint components.Subsequently,we simulated joint assembly buckling during bus rollover crash scenarios to verify and analyze the optimized solutions in three-point bending simulations.Each joint component showcased a remarkable 30%–40%mass reduction while boosting energy absorption.Our design optimization method exhibits high efficiency and costeffectiveness.Leveraging contemporary automation technology,the design optimization platform developed in this study is poised to facilitate intelligent optimization of lightweight metal components in future applications.

Effects of Temperature and Liquid Nitrogen(LN2)on Coal’s Mechanical and Acoustic Emission(AE)Properties

Liquid nitrogen has shown excellent performances as a good fracturing medium in the extraction of unconventional natural gas,and its application in coalbed methane extraction is currently a research hotspot.This study focuses on the acoustic emission properties of coal specimens treated utilizing liquid nitrogen with varying initial temperatures in a three-point bending environment.Through examination of the load-displacement curves of the considered coal samples,their mechanical properties are also revealed for different initial temperatures and cycling frequencies.The findings demonstrate a gradual decline in the maximum load capacity of coal rock as the temperature rises.Similarly,when subjected to the same temperature,an escalation in the cycling frequency leads to a reduction in the peak load of coal rock.This suggests that both temperature and cycling frequency exert a notable impact on the fracturing efficacy of liquid nitrogen.Freeze-thaw cycling treatments and exposure to high-temperature conditions can activate preexisting damage in the coal rock,and,accordingly,influence its mechanical properties.In particular,throughout the progressive loading of coal rock samples,the failure mechanisms are predominantly characterized by the occurrence of tensile cracks,succeeded by the development,spread,and fracture of shear fissures.

Three-point bending of honeycomb sandwich beams with facesheet perforations

A novel square honeycomb-cored sandwich beam with perforated bottom facesheet is investigated under threepoint bending,both analytically and numerically.Perforated square holes in the bottom facesheet are characterized by the area ratio of the hole to intact facesheet（perforation ratio）.While for large-scale engineering applications like the decks of cargo vehicles and transportation ships,the perforations are needed to facilitate the fabrication process（e.g.,laser welding）as well as service maintenance,it is demonstrated that these perforations,when properly designed,can also enhance the resistance of the sandwich to bending.For illustration,fair comparisons among competing sandwich designs having different perforation ratios but equal mass is achieved by systematically thickening the core webs.Further,the perforated sandwich beam is designed with a relatively thick facesheet to avoid local indention failure so that it mainly fails in two competing modes：（1）bending failure,i.e.,yielding of beam cross-section and buckling of top facesheet caused by bending moment;（2）shear failure,i.e.,yielding and buckling of core webs due to shear forcing.The sensitivity of the failure loads to the ratio of core height to beam span is also discussed for varying perforation ratios.As the perfo-ration ratio is increased,the load of shear failure increases due to thickening core webs,while that of bending failure decreases due to the weakening bottom facesheet.Design of a sandwich beam with optimal perforation ratio is realized when the two failure loads are equal,leading to significantly enhanced failure load（up to 60%increase）relative to that of a non-perforated sandwich beam with equal mass.

TWO POSITIVE SOLUTIONS TO THREE-POINT SINGULAR BOUNDARY VALUE PROBLEMS

In this article, we consider the existence of two positive solutions to nonlinear second order three-point singular boundary value problem： -u′′（t） = λf（t, u（t）） for all t ∈ （0, 1） subjecting to u（0） = 0 and αu（η） = u（1）, where η ∈ （0, 1）, α ∈ [0, 1）, and λ is a positive parameter. The nonlinear term f（t, u） is nonnegative, and may be singular at t = 0, t = 1, and u = 0. By the fixed point index theory and approximation method, we establish that there exists λ＊ ∈ （0, ＋∞], such that the above problem has at least two positive solutions for any λ ∈ （0, λ＊） under certain conditions on the nonlinear term f.

Existence of Sign-changing Solution for Three-point Boundary Value Problems

In this paper, by using the fixed-point index theory, we study the existence of sign-changing solution of some three-point boundary value problems {y ''(t) + f(y) = 0, t ∈ [0, 1], y' (0) = 0, y(1) = αy(η), where 0 < α < 1, 0 < η < 1, f : R → R is continuous, strictly increasing and f(0) = 0.

A CATASTROPHE ANALYSIS ON THE STABILITY OF THE CRACK GROWTH IN THREE-POINT BENDING SPECIMENS

This paper presents an attempt at the application of catastrophe theory to the stability analysis of J-controlled crack growth in three-point bending specimens. By introducing the solutions of J-integral in the completely yielding state for the ideal plastic material, the critical condition of losing stability for the crack propagation in the specimen is obtained, based on the cusp catastrophe theory. The process of the crack growth from geometrical sense is described.

On the Existence of Positive Solution for a Nonlinear Third-order Three-point Boundary Value Problem

An existence theorem of positive solution is established for a nonlinear third-order three-point boundary value problem. Here, we concentrated on the case that the nonlinear term is neither superlinear nor sublinear, and is not asymptotic at zero and infinity.

NONTRIVIAL SOLUTION OF A NONLINEAR SECOND-ORDER THREE-POINT BOUNDARY VALUE PROBLEM

In this paper, for a second-order three-point boundary value problem u″＋f（t,u）=0,0〈t〈1,au（0）-bu′（0）=0,u（1）-au（η）=0,where η∈ （0, 1）, a, b, α ∈R with a^2 ＋ b^2 〉 0, the existence of its nontrivial solution is studied. The＇conditions on f which guarantee the existence of nontrivial solution are formulated. As an application, some examples to demonstrate the results are given.

Positive solutions of three-point boundary value problems

In this paper, we consider existence of single or multiple positive solutions of three-point boundary value problems involving one-dimensional p-Laplacian. We then study existence of solutions when the problems are in resonance cases. The proposed approach is based on the Krasnoselskii＇s fixed point theorem and the coincidence degree.

Existence of Multiple Positive Solutions for Third-Order Three-Point Boundary Value Problem

In this paper, we study the existence of positive solutions for a class of third-order three-point boundary value problem. By employing the fixed point theorem on cone, some new criteria to ensure the three-point boundary value problem has at least three positive solutions are obtained. An example illustrating our main result is given. Moreover, some previous results will be improved significantly in our paper.

Positive Solutions of a Second-Order Three-point Boundary Value Problem

The existence of positive solutions is established for a nonlinear second-order three-point boundary value problem. The result improves and extends the main result in Electron J. Differential Equations, 34(1999), 1-8.

Three-point explicit compact difference scheme with arbitrary order of accuracy and its application in CFD

Based on the successive iteration in the Taylor series expansion method, a three-point explicit compact difference scheme with arbitrary order of accuracy is derived in this paper. Numerical characteristics of the scheme are studied by the Fourier analysisl Unlike the conventional compact difference schemes which need to solve the equation to obtain the unknown derivatives in each node, the proposed scheme is explicit and can achieve arbitrary order of accuracy in space. Application examples for the convectiondiffusion problem with a sharp front gradient and the typical lid-driven cavity flow are given. It is found that the proposed compact scheme is not only simple to implement and economical to use, but also is effective to simulate the convection-dominated problem and obtain high-order accurate solution in coarse grid systems.

Modelling and Simulation on the Effect of Hot Forming Damage on Three-Point Bending Performance of Beam Components

The effects of forming damage are analyzed,which occur during hot stamping process,on the load-carrying capacity and failure mode of hot stamped beams.A damage-coupled pre-forming constitutive model was proposed,in which the damage during hot stamping process was introduced into the service response.The constitutive model was applied into the three-point bending simulation of a hot stamped beam,and then the influences of forming damage on the load-carrying capacity and cracks propagation were investigated.The results show that the forming damage reduces the maximum load capacity of the hot stamped beam by 7.5%.It also causes the crack to occur earlier and promotes crack to propagate along the radial direction of the punch.

Experimental and theoretical analyses of package-on-package structure under three-point bending loading

High density packaging is developing toward miniaturization and integration, which causes many difficulties in designing, manufacturing, and reliability testing. Package-on-Package （POP） is a promising three-dimensional high- density packaging method that integrates a chip scale package （CSP） in the top package and a fine-pitch ball grid array （FBGA） in the bottom package. In this paper, in-situ scanning electron microscopy （SEM） observation is carried out to detect the deformation and damage of the PoP structure under three-point bending loading. The results indicate that the cracks occur in the die of the top package, then cause the crack deflection and bridging in the die attaching layer. Furthermore, the mechanical principles are used to analyse the cracking process of the PoP structure based on the multi-layer laminating hypothesis and the theoretical analysis results are found to be in good agreement with the experimental results.

Symmetric Positive Solutions of Nonlinear Singular Second-order Three-point Boundary Value Problem

In this paper, the second-order three-point boundary value problem u(t) + λa(t)f(t, u(t)) = 0, 0 < t < 1,u(t) = u(1- t), u(0)- u(1) = u(12)is studied, where λ is a positive parameter, under various assumption on a and f, we establish intervals of the parameter λ, which yield the existence of positive solution, our proof based on Krasnosel'skii fixed-point theorem in cone.{u"(t)+λa(t)f(t,u(t))=0,0<t<1,u(t)=u(1-t),u′(0)-u′(1)=u(1/2)is studied,where A is a positive parameter,under various assumption on a and f,we establish intervals of the parameter A,which yield the existence of positive solution,our proof based on Krasnosel'skii fixed-point theorem in cone.

Singular perturbation of a second-order three-point boundary value problem for nonlinear systems

This paper deals with the existence of solutions to a singularly perturbed second-order three-point boundary value problem for nonlinear differential systems. The authors construct an appropriate generalized lower- and upper-solution pair, a concept defined in this paper, and employ the Nagumo conditions and algebraic boundary layer functions to ensure the existence of solutions of the problem. The uniformly valid asymptotic estimate of the solutions is given as well. The differential systems have nonlinear dependence on all order derivatives of the unknown.

POSITIVE SOLUTIONS OF THREE-POINT BOUNDARY VALUE PROBLEM FOR SECOND ORDER DIFFERENTIAL EQUATIONS WITH AN ADVANCED ARGUMENT

By using the Krasnoselskii's fixed point theorem for cones,conditions for the existence of positive solutions to the three-point boundary value problem for second order differential equation with an advanced argumentu″(t)+λa(t)f(u(h(t)))=0, t∈(0,1), u(0)=0, αu(η)=u(1),where 0<η<1,0<α<1η and t≤h(t)≤1 are obtained.

Existence and Uniqueness of Positive Solution for Third-Order Three-Point Boundary Value Problems

This paper is devoted to the study of the existence and uniqueness of the positive solution for a type of the nonlinear third-order three-point boundary value problem. Our results are based on an iterative method and the Leray-Schauder fixed point theorem.

Monotone positive solution for three-point boundary value problem

In this paper, the existence of monotone positive solution for the following secondorder three-point boundary value problem is studied：x″（t）＋f（t,x（t））=0,0〈t〈1,x′（0）=0,x（1）＋δx′（η）=0,where η ∈ （0, 1）, δ∈ [0, ∞）, f ∈ C（[0, 1] × [0, ∞）, [0, ∞））. Under certain growth conditions on the nonlinear term f and by using a fixed point theorem of cone expansion and compression of functional type due to Avery, Anderson and Krueger, sufficient conditions for the existence of monotone positive solution are obtained and the bounds of solution are given. At last, an example is given to illustrate the result of the paper.