Nonsmooth Boundary Research Articles

Non-smooth Bayesian optimization in tuning scientific applications

Tuning algorithmic parameters to optimize the performance of large, complicated computational codes is an important problem involving finding the optima and identifying regimes defined by non-smooth boundaries in black-box functions. Within the Bayesian optimization framework, the Gaussian process surrogate model produces smooth mean functions, but functions in the tuning problem are often non-smooth, which is exacerbated by the fact that we usually have limited sequential samples from the black-box function. Motivated by these issues encountered in tuning, we propose a novel Gaussian process model called a clustered Gaussian process (cGP), where the components are dynamically updated by clustering. In our studies, the performance of cGP can be better than stationary GPs in nearly 90% of the experiments and better than non-stationary GPs in nearly 70% of the repeated experiments while requiring less computational cost. cGP provides a novel approach for dynamic GP, computes more efficiently than recursive partitioning, and discovers non-smoothness regimes. We provide extensive experiments including high-performance computing (HPC) and industrial simulation functions to show the effectiveness of our methods.

Jump relation for the conormal derivative of parabolic single layer potential

We consider a single layer potential generated by the fundamental solution of a second-order parabolic equation with Dini continuous coefficients. The potential is considered in a semibounded spatial domain with non-smooth lateral boundary from the Dini–Hölder class. We establish a jump relation for the conormal derivative of this potential at the lateral boundary of the domain.

Accurate and efficient boundary method for isogeometric shape design sensitivity analysis considering tangential divergence of non-smooth boundary

Accurate and efficient boundary method for isogeometric shape design sensitivity analysis considering tangential divergence of non-smooth boundary

Singularly perturbed elliptic problems of convection–diffusion type with non-smooth inflow/outflow boundary conditions

A singularly perturbed elliptic problem, with non-smooth inflow or non-smooth outflow boundary conditions, is examined. A decomposition of the continuous solution is constructed, whose components identify the various types of layer functions that can exist in the solution. Parameter-explicit pointwise bounds on the first three partial derivatives of these components are established. An appropriate Shishkin mesh is identified for the problem and this is combined with upwinding to form a numerical method. Parameter-uniform error bounds are deduced. Numerical results are presented to illustrate the performance of the numerical method.

On well-posedness of Navier–Stokes variational inequalities

In this paper, well-posedness is carried out on variational inequalities in a general form for the stationary Navier–Stokes equations modeling viscous incompressible fluid flows with a nonsmooth slip boundary condition of friction type. For a special form of the variational inequality commonly studied in existing literature, our proof technique leads to a substantial improvement on the well-posedness condition.

On a Stokes hemivariational inequality for incompressible fluid flows with damping

In this paper, a Stokes hemivariational inequality is studied for incompressible fluid flows with the damping effect. The hemivariational inequality feature is caused by the presence of a nonsmooth slip boundary condition of friction type. Well-posedness of the Stokes hemivariational inequality is established through the consideration of a minimization problem. Mixed finite element methods are introduced to solve the Stokes hemivariational inequality and error estimates are derived for the mixed finite element solutions. The error estimates are of optimal order for low-order mixed element pairs under suitable solution regularity assumptions. An efficient iterative algorithm is introduced to solve the mixed finite element system. Numerical results are reported on the performance of the proposed algorithm and the numerical convergence orders of the finite element solutions.

A NEW PERSPECTIVE ON THE NONLINEAR DATE–JIMBO–KASHIWARA–MIWA EQUATION IN FRACTAL MEDIA

In this paper, we first created a fractal Date–Jimbo–Kashiwara–Miwa (FDJKM) long ripple wave model in a non-smooth boundary or microgravity space recorded. Using fractal semi-inverse skill (FSIS) and fractal traveling wave transformation (FTWT), the fractal variational principle (FVP) was derived, and the strong minimum necessary circumstance was attested with the He Wierstrass function. We have discovered two distinct solitary wave solutions, the square form of the hyperbolic secant function and the hyperbolic secant function form. Then, soliton solutions are cultivated through FVP and the minimum steady state condition. Finally, the influences of non-smooth boundaries on solitons were tackled, and the properties of the solution were demonstrated through three-dimensional contour lines. Fractal dimension can impact waveforms, but cannot affect their vertex values. The presentation of soliton solutions (SWS) using techniques is not only laudable but also noteworthy. The technique employed can also be used to investigate solitary wave solutions of other local fractional calculus partial differential equations.

Solitary wave solution for the non-linear bending wave equation based on He’s variational method

A beam vibration originating in the beam porous structure or on a non-smooth boundary might make its vibrating energy concentrated on a single wave, leading to a solitary wave. This paper applies the variational approach to analysis of the soliton basic property, and the effect of the fractal dimensions on the solitary wave is elucidated. This paper is to draw attention the beam soliton property be-yond its widely known resonance and periodic and chaotic properties.

ASYMPTOTICS OF THE SOLUTION OF PARABOLIC PROBLEMS WITH NONSMOOTH BOUNDARY FUNCTIONS

ASYMPTOTICS OF THE SOLUTION OF PARABOLIC PROBLEMS WITH NONSMOOTH BOUNDARY FUNCTIONS

Fast-slow dynamics analysis in an externally excited smooth and discontinuous oscillator with a pair of irrational nonlinearities

The study of fast-slow oscillations in systems with irrational nonlinearity that may yield abundant dynamical mechanisms is not well developed. This paper aims to investigate the fast-slow dynamics in an excited mass-spring oscillator with a pair of irrational nonlinearities, which can undergo the dynamical transition from smooth to discontinuous characteristics depending on the values of a smoothness parameter. Three different types of fast-slow oscillations are reported in this interesting smooth and discontinuous (SD) oscillator with a pair of irrational nonlinearities. Due to the smooth and discontinuous characteristics of this SD oscillator, we consider its dynamical behaviors under the smooth and discontinuous cases, respectively. Based on the fast-slow analysis and the two-parameter bifurcation analysis, the smooth fast-slow dynamics associated with fold hysteresis and its turnover are revealed. In the discontinuous case, the system can be viewed as a piecewise-smooth dynamical system governed by three different subsystems in different regions divided by two nonsmooth boundaries. In particular, the nonsmooth boundaries can be divided into parts with different dynamical behaviors, including escaping and crossing lines. Unlike the smooth case, there is no change in the stability of the equilibrium in these three subsystems. However, transitions of system trajectory induced by crossing lines can account for the generation of fast-slow oscillations in the piecewise-smooth system. As a result, the smooth and piecewise-smooth fast-slow dynamics in the excited SD oscillator with a pair of irrational nonlinearities are revealed, which deepens the understanding of fast-slow dynamics of the dynamical systems with irrational nonlinearity.

Nachal'no-kraevye zadachi dlya odnorodnykh parabolicheskikh sistem v poluogranichennoy ploskoy oblasti i uslovie dopolnitel'nosti

We consider initial–boundary value problems for homogeneous parabolic systemswith coefficients satisfying the double Dini condition with zero initial conditions in a semiboundedplane domain with nonsmooth lateral boundary. The method of boundary integralequations is used to prove a theorem on the unique classical solvability of such problems in thespace of functions that are continuous together with their first spatial derivative in the closureof the domain. An integral representation of the obtained solutions is given. It is shown thatthe condition for the solvability of the posed problems considered in the paper is equivalent tothe well-known complementarity condition.

Sushchestvovanie i edinstvennost' klassicheskogo resheniya pervoy kraevoy zadachi dlya parabolicheskikh sistem na ploskosti

We consider the first boundary value problem for uniformly parabolic systems of the second order with one spatial variable in bounded and semibounded domains with nonsmooth lateral boundaries. The coefficients of the system satisfy the Hölder condition and do not depend on the time variable. For continuous initial and boundary functions, the existence and uniqueness of the classical solution of this problem is established

Diffusion in media with membranes and some nonlocal parabolic problems

UDC 519.21 We establish the classical solvability of a certain conjugation problem for one-dimensional (with respect to a spatial variable) Kolmogorov backward equation with discontinuous coefficients and some variants of the general nonlocal Feller–Wentzell boundary condition given on nonsmooth boundaries of considered curvilinear domains. In addition, we prove, that the two-parameter Feller semigroup defined by the solution of this problem describes some inhomogeneous diffusion process with moving membranes on the given region of the real line. We also show the relationship between the constructed process and the generalized diffusion in the sense of M. I. Portenko.

Identification of discontinuous parameters in contaminant convection–reaction–diffusion model of recovered fracturing fluid

The primary goal of this paper is to identify the discontinuous parameters in a complicated contaminant convection–reaction–diffusion model of recovered fracturing fluid (RFF model, for short), which is modeled by a nonlinear stationary incompressible Navier–Stokes equation with multivalued and nonsmooth friction boundary condition coupled with a nonlinear convection–reaction–diffusion equation involving mixed Neumann boundary conditions. First, we impose the parameter identification problem which is, exactly, a nonlinear and nonsmooth inverse problem. Then, we show the local boundedness and weakly relative compactness of solution map to RFF model with respect to the discontinuous parameters. Moreover, the generalized continuity of solution map of RFF model is proved. Finally, via employing the theory of nonsmooth analysis and optimization, a sufficiency theorem of existence of solutions to the inverse problem under consideration is established.

Multi-patch isogeometric material optimization of bi-directional functionally graded plates

This study investigates an effective design methodology for the optimal material distribution of bi-directional functionally graded plates (2D-FGPs) with complex shape. In mechanical analysis, a multi-patch isogeometric method is used to analyze the statics of 2D-FGP, which is based on the third-order shear deformation theory and Nitsche’s technology. This method can effectively avoid the use of duplicate nodes in the parameterization of IGA for complex shapes to obtain C1-continuity of 2D-FGPs. In the optimal design problem, we have constructed a rectangular material design mesh (RMDM) based on the shape of 2D-FGPs, which can map the material distribution on the surface of plate to achieve the implementation of the optimization process. This two-dimensional B-spline control points of RMDM are set as the design variable with mass reduction and the first-order natural frequency maximization as optimization objectives, and limited arbitrary deflection as constraint conditions. In addition, an improved multi-objective particle swarm optimization algorithm (IMOPSO) is used to obtain a series of Pareto optimization solutions that meet the needs of the designer. The validity and applicability of this innovative combination of multi-patch isogeometric analysis and IMOPSO are demonstrated through several numerical examples in integrated design. This approach further accomplishes the numerical unified CAD/CAE optimization design of 2D-FGPs across multiple non-smooth boundaries.

Swgan: A new algorithm of adhesive rice image segmentation based on improved generative adversarial networks

Image segmentation is a crucial part in the automatic detection of rice appearance quality. Due to morphological characteristics of rice grains, missed detection and non-smooth boundaries may exist in the image segmentation of adhesive rice. To address the above issues, this study proposes a novel model named Swgan combined generative adversarial networks (GANs) with nested skip connections for obtaining accurate masks. In order to learn the mask distribution of each object in adhesive rice image and further avoid missed detection, the discriminator in GAN is used as a modifier of Cascade Mask R-CNN model to allow the generator to overcome the limitation of mask generation training, namely detecting multiply objects as a single target. Moreover, the Swgan utilizes Swin-Transformer as the backbone network and incorporates the Cascade Mask R-CNN framework and Nested Feature Pyramid Network (Nested-FPN) to maintain the mask’s boundary smoothness during forward propagation. Experimental results indicate that the Swgan is used to obtain better segmentation results from objective detection and segmentation under complex conditions of adhesive rice when compared with state-of-art algorithms. Overall, the Swgan with satisfactory accuracy in image segmentation of adhesive rice combined with physical indicators detection provide reliable quality assessment of rice grains.

Semianalytical Solution for Dissipation Process of Partially Saturated Soils Considering Nonsmooth Boundary and Stress Level

Semianalytical Solution for Dissipation Process of Partially Saturated Soils Considering Nonsmooth Boundary and Stress Level

A New Definition of the Dual Interpolation Curve for CAD Modeling and Geometry Defeaturing

The present paper provides a new definition of the dual interpolation curve in a geometric-intuitive way based on adaptive curve refinement techniques. The dual interpolation curve is an implementation of the interpolatory subdivision scheme for curve modeling, which comprises polynomial segments of different degrees. Dual interpolation curves maintain various desirable properties of conventional curve modeling methods, such as local adaptive subdivision, high interpolation accuracy and convergence, and continuous and discontinuous boundary representation. In addition, the dual interpolation curve is mainly applied to solve the difficult geometry defeaturing problems for curve modeling in existing computer-aided technology. By adding fictitious and intrinsic nodes inside or at the vertices of interpolation elements, the dual interpolation curve is flexible and convenient for characterizing a set of ordered points or discrete segments. Combined with the Lagrange interpolation polynomial and meshless method, the proposed approach is capable of characterizing the non-smooth boundary for geometry defeaturing. Experimental results are given to verify the validity, robustness, and accuracy of the proposed method.

A high-order embedded-boundary method based on smooth extension and RBFs for solving elliptic equations in multiply connected domains

This paper presents a new high-order embedded-/immersed-boundary method, based on point collocation, smooth extension of the solution and integrated radial basis functions (IRBFs), for solving the elliptic partial differential equation (PDE) defined in a domain with holes. The PDE is solved in the domain without holes, where the construction of the IRBF approximations is based on a fixed Cartesian grid and local five-point stencils, and the inner/immersed boundary conditions are included in the discretized equations. More importantly, nodal values of the second-, third- and fourth-order derivatives of the field variable are incorporated into the IRBF approximations, and the forcing term defined in the holes is constructed in a form that gives a globally smooth solution. These features enable the proposed scheme to achieve high level of sparseness of the system matrix, and high level of accuracy of the solution together. Numerical verification is carried out for problems with smooth and non-smooth inner boundaries. Highly accurate results are obtained using relatively coarse grids.

Three-dimensional mapping analysis of a capsule system with bilateral elastic constraints

The three-dimensional mapping of the capsule system with bilateral elastic constraints is studied, which is rarely seen in the literature at present. The bilaterally constrained elastic system is suitable for small-scale capsule models. Compared with the unilaterally constrained elastic capsule system, the bilaterally constrained elastic system has a more complex switching mechanism on the boundary, a larger number of local mappings, and a richer whole motion process and bifurcation. The detailed analysis of the bilaterally constrained elastic capsule models can provide more effective theoretical support for the practical application of small-scale capsule models. In this paper, the local dynamics of non-smooth boundaries caused by dry friction and contact and global dynamics in a bilaterally constrained elastic capsule system are analyzed. By considering the vector fields near the boundary surfaces and boundary lines, we get the switching mechanisms on the boundary surfaces and boundary lines and find all possible local mappings. Under the guidance of theoretical analysis, we obtain the sliding bifurcations and boundary-intersection crossing bifurcations by changing the external excitation amplitude. The numerical simulation shows that the bifurcation points and the change of the motion state before and after the bifurcation are consistent with the theoretical results, indicating that this analysis method can be used to accurately discuss the dynamic behaviors of the bilaterally constrained elastic capsule system and mappings can express the dynamic behaviors.