Dirac method for nonlinear and non-homogenous boundary value problems of plates

The boundary value problem plays a crucial role in the analytical investigation of continuum dynamics. In this paper, an analytical method based on the Dirac operator to solve the nonlinear and non-homogeneous boundary value problem of rectangular plates is proposed. The key concept behind this method is to transform the nonlinear or non-homogeneous part on the boundary into a lateral force within the governing function by the Dirac operator, which linearizes and homogenizes the original boundary, allowing one to employ the modal superposition method for obtaining solutions to reconstructive governing equations. Once projected into the modal space, the harmonic balance method(HBM) is utilized to solve coupled ordinary differential equations(ODEs)of truncated systems with nonlinearity. To validate the convergence and accuracy of the proposed Dirac method, the results of typical examples, involving nonlinearly restricted boundaries, moment excitation, and displacement excitation, are compared with those of the differential quadrature element method(DQEM). The results demonstrate that when dealing with nonlinear boundaries, the Dirac method exhibits more excellent accuracy and convergence compared with the DQEM. However, when facing displacement excitation, there exist some discrepancies between the proposed approach and simulations;nevertheless, the proposed method still accurately predicts resonant frequencies while being uniquely capable of handling nonuniform displacement excitations. Overall, this methodology offers a convenient way for addressing nonlinear and non-homogenous plate boundaries.

Grain boundary engineering for enhancing intergranular damage resistance of ferritic/martensitic steel P92

Ferritic/martensitic(F/M)steel is widely used as a structural material in thermal and nuclear power plants.However,it is susceptible to intergranular damage,which is a critical issue,under service conditions.In this study,to improve the resistance to intergranular damage of F/M steel,a thermomechanical process(TMP)was employed to achieve a grain boundary engineering(GBE)microstructure in F/M steel P92.The TMP,including cold-rolling thickness reduction of 6%,9%,and 12%,followed by austenitization at 1323 K for 40 min and tempering at 1053 K for 45 min,was applied to the as-received(AR)P92 steel.The prior austenite grain(PAG)size,prior austenite grain boundary character distribution(GBCD),and connectivity of prior austenite grain boundaries(PAGBs)were investigated.Compared to the AR specimen,the PAG size did not change significantly.The fraction of coincident site lattice boundaries(CSLBs,3≤Σ≤29)and Σ3^(n) boundaries along PAGBs decreased with increasing reduction ratio because the recrystallization fraction increased with increasing reduction ratio.The PAGB connectivity of the 6%deformed specimen slightly deteriorated compared with that of the AR specimen.Moreover,potentiodynamic polarization studies revealed that the intergranular damage resistance of the studied steel could be improved by increasing the fraction of CSLBs along the PAGBs,indicating that the TMP,which involves low deformation,could enhance the intergranular damage resistance.

Wavelet Multi-Resolution Interpolation Galerkin Method for Linear Singularly Perturbed Boundary Value Problems

In this study,a wavelet multi-resolution interpolation Galerkin method(WMIGM)is proposed to solve linear singularly perturbed boundary value problems.Unlike conventional wavelet schemes,the proposed algorithm can be readily extended to special node generation techniques,such as the Shishkin node.Such a wavelet method allows a high degree of local refinement of the nodal distribution to efficiently capture localized steep gradients.All the shape functions possess the Kronecker delta property,making the imposition of boundary conditions as easy as that in the finite element method.Four numerical examples are studied to demonstrate the validity and accuracy of the proposedwavelet method.The results showthat the use ofmodified Shishkin nodes can significantly reduce numerical oscillation near the boundary layer.Compared with many other methods,the proposed method possesses satisfactory accuracy and efficiency.The theoretical and numerical results demonstrate that the order of theε-uniform convergence of this wavelet method can reach 5.

A High-Accuracy Curve Boundary Recognition Method Based on the Lattice Boltzmann Method and Immersed Moving Boundary Method

Applying numerical simulation technology to investigate fluid-solid interaction involving complex curved bound-aries is vital in aircraft design,ocean,and construction engineering.However,current methods such as Lattice Boltzmann(LBM)and the immersion boundary method based on solid ratio(IMB)have limitations in identifying custom curved boundaries.Meanwhile,IBM based on velocity correction(IBM-VC)suffers from inaccuracies and numerical instability.Therefore,this study introduces a high-accuracy curve boundary recognition method(IMB-CB),which identifies boundary nodes by moving the search box,and corrects the weighting function in LBM by calculating the solid ratio of the boundary nodes,achieving accurate recognition of custom curve boundaries.In addition,curve boundary image and dot methods are utilized to verify IMB-CB.The findings revealed that IMB-CB can accurately identify the boundary,showing an error of less than 1.8%with 500 lattices.Also,the flow in the custom curve boundary and aerodynamic characteristics of the NACA0012 airfoil are calculated and compared to IBM-VC.Results showed that IMB-CB yields lower lift and drag coefficient errors than IBM-VC,with a 1.45%drag coefficient error.In addition,the characteristic curve of IMB-CB is very stable,whereas that of IBM-VC is not.For the moving boundary problem,LBM-IMB-CB with discrete element method(DEM)is capable of accurately simulating the physical phenomena of multi-moving particle flow in complex curved pipelines.This research proposes a new curve boundary recognition method,which can significantly promote the stability and accuracy of fluid-solid interaction simulations and thus has huge applications in engineering.

Two Stages Segmentation Algorithm of Breast Tumor in DCE-MRI Based on Multi-Scale Feature and Boundary Attention Mechanism

Nuclearmagnetic resonance imaging of breasts often presents complex backgrounds.Breast tumors exhibit varying sizes,uneven intensity,and indistinct boundaries.These characteristics can lead to challenges such as low accuracy and incorrect segmentation during tumor segmentation.Thus,we propose a two-stage breast tumor segmentation method leveraging multi-scale features and boundary attention mechanisms.Initially,the breast region of interest is extracted to isolate the breast area from surrounding tissues and organs.Subsequently,we devise a fusion network incorporatingmulti-scale features and boundary attentionmechanisms for breast tumor segmentation.We incorporate multi-scale parallel dilated convolution modules into the network,enhancing its capability to segment tumors of various sizes through multi-scale convolution and novel fusion techniques.Additionally,attention and boundary detection modules are included to augment the network’s capacity to locate tumors by capturing nonlocal dependencies in both spatial and channel domains.Furthermore,a hybrid loss function with boundary weight is employed to address sample class imbalance issues and enhance the network’s boundary maintenance capability through additional loss.Themethod was evaluated using breast data from 207 patients at RuijinHospital,resulting in a 6.64%increase in Dice similarity coefficient compared to the benchmarkU-Net.Experimental results demonstrate the superiority of the method over other segmentation techniques,with fewer model parameters.

The Regularity of Solutions to Mixed Boundary Value Problems of Second-Order Elliptic Equations with Small Angles

This paper considers the regularity of solutions to mixed boundary value problems in small-angle regions for elliptic equations. By constructing a specific barrier function, we proved that under the assumption of sufficient regularity of boundary conditions and coefficients, as long as the angle is sufficiently small, the regularity of the solution to the mixed boundary value problem of the second-order elliptic equation can reach any order.

GLOBAL SOLUTIONS TO 1D COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND FREE-BOUNDARY

This paper is concerned with the Navier-Stokes/Allen-Cahn system,which is used to model the dynamics of immiscible two-phase flows.We consider a 1D free boundary problem and assume that the viscosity coefficient depends on the density in the form ofη(ρ)=ρ^(α).The existence of unique global H^(2m)-solutions(m∈N)to the free boundary problem is proven for when 0<α<1/4.Furthermore,we obtain the global C^(∞)-solutions if the initial data is smooth.

Dynamics of fundamental and double-pole breathers and solitons for a nonlinear Schr¨odinger equation with sextic operator under non-zero boundary conditions

We study the dynamics of fundamental and double-pole breathers and solitons for the focusing and defocusing nonlinearSchr¨odinger equation with the sextic operator under non-zero boundary conditions. Our analysis mainly focuses onthe dynamical properties of simple- and double-pole solutions. Firstly, through verification, we find that solutions undernon-zero boundary conditions can be transformed into solutions under zero boundary conditions, whether in simple-pole ordouble-pole cases. For the focusing case, in the investigation of simple-pole solutions, temporal periodic breather and thespatial-temporal periodic breather are obtained by modulating parameters. Additionally, in the case of multi-pole solitons,we analyze parallel-state solitons, bound-state solitons, and intersecting solitons, providing a brief analysis of their interactions.In the double-pole case, we observe that the two solitons undergo two interactions, resulting in a distinctive “triangle”crest. Furthermore, for the defocusing case, we briefly consider two situations of simple-pole solutions, obtaining one andtwo dark solitons.

Riemann–Hilbert problem for the defocusing Lakshmanan–Porsezian–Daniel equation with fully asymmetric nonzero boundary conditions

The Riemann–Hilbert approach is demonstrated to investigate the defocusing Lakshmanan–Porsezian–Daniel equation under fully asymmetric nonzero boundary conditions.In contrast to the symmetry case,this paper focuses on the branch points related to the scattering problem rather than using the Riemann surfaces.For the direct problem,we analyze the Jost solution of lax pairs and some properties of scattering matrix,including two kinds of symmetries.The inverse problem at branch points can be presented,corresponding to the associated Riemann–Hilbert.Moreover,we investigate the time evolution problem and estimate the value of solving the solutions by Jost function.For the inverse problem,we construct it as a Riemann–Hilbert problem and formulate the reconstruction formula for the defocusing Lakshmanan–Porsezian–Daniel equation.The solutions of the Riemann–Hilbert problem can be constructed by estimating the solutions.Finally,we work out the solutions under fully asymmetric nonzero boundary conditions precisely via utilizing the Sokhotski–Plemelj formula and the square of the negative column transformation with the assistance of Riemann surfaces.These results are valuable for understanding physical phenomena and developing further applications of optical problems.

CONVEXITY OF THE FREE BOUNDARY FOR AN AXISYMMETRIC INCOMPRESSIBLE IMPINGING JET

This paper is devoted to the study of the shape of the free boundary for a threedimensional axisymmetric incompressible impinging jet.To be more precise,we will show that the free boundary is convex to the fluid,provided that the uneven ground is concave to the fluid.

Experimental investigation on the permeability of gap-graded soil due to horizontal suffusion considering boundary effect

The boundary condition is a crucial factor affecting the permeability variation due to suffusion.An experimental investigation on the permeability of gap-graded soil due to horizontal suffusion considering the boundary effect is conducted,where the hydraulic head difference(DH)varies,and the boundary includes non-loss and soil-loss conditions.Soil samples are filled into seven soil storerooms connected in turn.After evaluation,the variation in content of fine sand(ΔR_(f))and the hydraulic conductivity of soils in each storeroom(C_(i))are analyzed.In the non-loss test,the soil sample filling area is divided into runoff,transited,and accumulated areas according to the negative or positive ΔR_(f) values.ΔR_(f) increases from negative to positive along the seepage path,and Ci decreases from runoff area to transited area and then rebounds in accumulated area.In the soil-loss test,all soil sample filling areas belong to the runoff area,where the gentle-loss,strengthened-loss,and alleviated-loss parts are further divided.ΔR_(f) decreases from the gentle-loss part to the strengthened-loss part and then rebounds in the alleviated-loss part,and C_(i) increases and then decreases along the seepage path.The relationship between ΔR_(f) and Ci is different with the boundary condition.Ci exponentially decreases with ΔR_(f) in the non-loss test and increases with ΔR_(f) generally in the soil-loss test.

Delineation of urban growth boundary based on FLUS model under the perspective of land use evaluation in hilly mountainous areas

With rapid economic development,the size of urban land in China is expanding dramatically.The Urban Growth Boundary(UGB)is an expandable spatial boundary for urban construction in a certain period in order to control the urban sprawl.Reasonable delineation of UGB can inhibit the disorderly spread of urban space and guide the normal development of the city.It is of practical significance for the construction of green urban space.The study utilizes GIS technology to establish a land construction suitability evaluation system for Nankang city,which is experiencing rapid urban expansion,and outlines the preliminary UGB under the future land use simulation(FLUS)model.At the same time,considering the coupled coordination of"Production-Living-Ecological Space",and based on the suitability evaluation,we revised the preliminary UGB by combining the advantages of the patch-generating land use simulation(PLUS)model and the convex hull model to delineate the final UGB.The results show that:1)the comprehensive score of the evaluation of the suitability of the construction of land from high to low shows the distribution of the center of the city to the surrounding circle type spread,the center of the city has the highest suitability score.The results of convex hull model show that the urban expansion type of Nankang is epitaxial.In the future,the urban expansion will mainly occur in the northern part of the city.The PLUS model predicts an increase of 3359.97 hm^(2)of construction land in Nankang by 2035,of which 2022.97 hm^(2)is urban construction land.2)The FLUS model has a prediction accuracy of 86.3%and delineates a preliminary UGB area of 9215.07 hm^(2).3)We used the results of the construction suitability evaluation,PLUS model simulation results,and convex hull model predictions to revise the originally delineated UGB.The final delineated UGB area is 8895.67 hm^(2)and it is capable of meeting the future development of the study area.The results of the delineation can promote sustainable urban development,and the delineation methodology can provide a reference basis for the preparation of territorial spatial planning.

Atomistic study on the microscopic mechanism of grain boundary embrittlement induced by small dense helium bubbles in iron

The helium bubbles induced by 14 MeV neutron irradiation can cause intergranular fractures in reduced activation ferritic martensitic steel,which is a candidate structural material for fusion reactors.In order to elucidate the susceptibility of different grain boundaries(GBs)to helium-induced embrittlement,the tensile fracture processes of 10 types of GBs with and without helium bubbles in body-centered cubic(bcc)iron at the relevant service temperature of 600 K were investigated via molecular dynamics methods.The results indicate that in the absence of helium bubbles,the GBs studied here can be classified into two distinct categories:brittle GBs and ductile GBs.The atomic scale analysis shows that the plastic deformation of ductile GB at high temperatures originates from complex plastic deformation mechanisms,including the Bain/Burgers path phase transition and deformation twinning,in which the Bain path phase transition is the most dominant plastic deformation mechanism.However,the presence of helium bubbles severely inhibits the plastic deformation channels of the GBs,resulting in a significant decrease in elongation at fractures.For bubble-decorated GBs,the ultimate tensile strength increases with the increase in the misorientation angle.Interestingly,the coherent twin boundary∑3{112}was found to maintain relatively high fracture strength and maximum failure strain under the influence of helium bubbles.

CAUCHY TYPE INTEGRALS AND A BOUNDARY VALUE PROBLEM IN A COMPLEX CLIFFORD ANALYSIS

Clifford analysis is an important branch of modern analysis;it has a very important theoretical significance and application value,and its conclusions can be applied to the Maxwell equation,Yang-Mill field theory,quantum mechanics and value problems.In this paper,we first give the definition of a quasi-Cauchy type integral in complex Clifford analysis,and get the Plemelj formula for it.Second,we discuss the H?lder continuity for the Cauchy-type integral operators with values in a complex Clifford algebra.Finally,we prove the existence of solutions for a class of linear boundary value problems and give the integral representation for the solution.

Towards designing high mechanical performance low-alloyed wrought magnesium alloys via grain boundary segregation strategy:A review

Low-alloyed magnesium(Mg)alloys have emerged as one of the most promising candidates for lightweight materials.However,their further application potential has been hampered by limitations such as low strength,poor plasticity at room temperature,and unsatisfactory formability.To address these challenges,grain refinement and grain structure control have been identified as crucial factors to achieving high performance in low-alloyed Mg alloys.An effective way for regulating grain structure is through grain boundary(GB)segregation.This review presents a comprehensive summary of the distribution criteria of segregated atoms and the effects of solute segregation on grain size and growth in Mg alloys.The analysis encompasses both single element segregation and multi-element co-segregation behavior,considering coherent interfaces and incoherent interfaces.Furthermore,we introduce the high mechanical performance low-alloyed wrought Mg alloys that utilize GB segregation and analyze the potential impact mechanisms through which GB segregation influences materials properties.Drawing upon these studies,we propose strategies for the design of high mechanical performance Mg alloys with desirable properties,including high strength,excellent ductility,and good formability,achieved through the implementation of GB segregation.The findings of this review contribute to advancing the understanding of grain boundary engineering in Mg alloys and provide valuable insights for future alloy design and optimization.

Multi-soliton solutions of coupled Lakshmanan-Porsezian-Daniel equations with variable coefficients under nonzero boundary conditions

This paper aims to investigate the multi-soliton solutions of the coupled Lakshmanan–Porsezian–Daniel equations with variable coefficients under nonzero boundary conditions.These equations are utilized to model the phenomenon of nonlinear waves propagating simultaneously in non-uniform optical fibers.By analyzing the Lax pair and the Riemann–Hilbert problem,we aim to provide a comprehensive understanding of the dynamics and interactions of solitons of this system.Furthermore,we study the impacts of group velocity dispersion or the fourth-order dispersion on soliton behaviors.Through appropriate parameter selections,we observe various nonlinear phenomena,including the disappearance of solitons after interaction and their transformation into breather-like solitons,as well as the propagation of breathers with variable periodicity and interactions between solitons with variable periodicities.

Development of D_α band symmetrical visible optical diagnostic for boundary reconstruction on EAST tokamak

To investigate the potential of utilizing visible spectral imaging for controlling the plasma boundary shape during stable operation of plasma in future tokamak, a D_α band symmetric visible light diagnostic system was designed and implemented on the Experimental Advanced Superconducting Tokamak(EAST). This system leverages two symmetric optics for joint plasma imaging. The optical system exhibits a spatial resolution less than 2 mm at the poloidal cross-section, distortion within the field of view below 10%, and relative illumination of 91%.The high-quality images obtained enable clear observation of both the plasma boundary position and the characteristics of components within the vacuum vessel. Following system calibration and coordinate transformation, the image coordinate boundary features are mapped to the tokamak coordinate system. Utilizing this system, the plasma boundary was reconstructed, and the resulting representation showed alignment with the EFIT(Equilibrium Fitting) results. This underscores the system's superior performance in boundary reconstruction applications and provides a diagnostic foundation for boundary shape control based on visible spectral imaging.

THE INTERIOR TRANSMISSION EIGENVALUE PROBLEM FOR AN ANISOTROPIC MEDIUM BY A PARTIALLY COATED BOUNDARY

We consider the interior transmission eigenvalue problem corresponding to the scattering for an anisotropic medium of the scalar Helmholtz equation in the case where the boundary?Ωis split into two disjoint parts and possesses different transmission conditions.Using the variational method,we obtain the well posedness of the interior transmission problem,which plays an important role in the proof of the discreteness of eigenvalues.Then we achieve the existence of an infinite discrete set of transmission eigenvalues provided that n≡1,where a fourth order differential operator is applied.In the case of n■1,we show the discreteness of the transmission eigenvalues under restrictive assumptions by the analytic Fredholm theory and the T-coercive method.

Analytical solutions of turbulent boundary layer beneath forward-leaning waves

As a typical nonlinear wave,forward-leaning waves can be frequently encountered in the near-shore areas,which can impact coastal sediment transport significantly.Hence,it is of significance to describe the characteristics of the boundary layer beneath forward-leaning waves accurately,especially for the turbulent boundary layer.In this work,the linearized turbulent boundary layer model with a linear turbulent viscosity coefficient is applied,and the novel expression of the near-bed orbital velocity that has been worked out by the authors for forward-leaning waves of arbitrary forward-leaning degrees is further used to specify the free stream boundary condition of the bottom boundary layer.Then,a variable transformation is found so as to make the equation of the turbulent boundary layer model be solved analytically through a modified Bessel function.Consequently,an explicit analytical solution of the turbulent boundary layer beneath forward-leaning waves is derived by means of variable separation and variable transformation.The analytical solutions of the velocity profile and bottom shear stress of the turbulent boundary layer beneath forward-leaning waves are verified by comparing the present analytical results with typical experimental data available in the previous literature.

Effect of boundary conditions on shakedown analysis of heterogeneous materials

The determination of the ultimate load-bearing capacity of structures made of elastoplastic heterogeneous materials under varying loads is of great importance for engineering analysis and design. Therefore, it is necessary to accurately predict the shakedown domains of these materials. The static shakedown theorem, also known as Melan's theorem, is a fundamental method used to predict the shakedown domains of structures and materials. Within this method, a key aspect lies in the construction and application of an appropriate self-equilibrium stress field(SSF). In the structural shakedown analysis, the SSF is typically constructed by governing equations that satisfy no external force(NEF) boundary conditions. However, we discover that directly applying these governing equations is not suitable for the shakedown analysis of heterogeneous materials. Researchers must consider the requirements imposed by the Hill-Mandel condition for boundary conditions and the physical significance of representative volume elements(RVEs). This paper addresses this issue and demonstrates that the sizes of SSFs vary under different boundary conditions, such as uniform displacement boundary conditions(DBCs), uniform traction boundary conditions(TBCs), and periodic boundary conditions(PBCs). As a result, significant discrepancies arise in the predicted shakedown domain sizes of heterogeneous materials. Built on the demonstrated relationship between SSFs under different boundary conditions, this study explores the conservative relationships among different shakedown domains, and provides proof of the relationship between the elastic limit(EL) factors and the shakedown loading factors under the loading domain of two load vertices. By utilizing numerical examples, we highlight the conservatism present in certain results reported in the existing literature. Among the investigated boundary conditions, the obtained shakedown domain is the most conservative under TBCs.Conversely, utilizing PBCs to construct an SSF for the shakedown analysis leads to less conservative lower bounds, indicating that PBCs should be employed as the preferred boundary conditions for the shakedown analysis of heterogeneous materials.