Ricci flow on manifolds with boundary with arbitrary initial metric

In this paper, we study the Ricci flow on manifolds with boundary. In the paper, we substantially improve Shen’s result [Y. Shen, On Ricci deformation of a Riemannian metric on manifold with boundary, Pacific J. Math. 173 1996, 1, 203–221] to manifolds with arbitrary initial metric. We prove short-time existence and uniqueness of the solution, in which the boundary becomes instantaneously totally geodesic for positive time. Moreover, we prove that the flow we constructed preserves natural boundary conditions. More specifically, if the initial metric has a convex boundary, then the flow preserves positive curvature operator and the PIC1, PIC2 conditions. Moreover, if the initial metric has a two-convex boundary, then the flow preserves the PIC condition.

On outflow boundary conditions in finite element simulations of non-Newtonian internal flows

The matter of appropriate boundary conditions for open or truncated outflow regions in internal flow is still focus of discussion and research. In most practical applications, one can at best estimate mean pressure values or flow rates at such outlets. In the context of finite element methods, it is known that enforcing mean pressures through the pseudo-tractions arising from the Laplacian Navier-Stokes formulation yields accurate, physically consistent solutions. Nevertheless, when generalised Newtonian fluid models are considered, the resulting non-uniform viscosity fields render the classical Laplacian formulation inadequate. Thus, it is common practice to use the socalled stress-divergence formulation with natural boundary conditions known for causing nonphysical outflow behaviour. In order to overcome such a limitation, this work presents a novel mixed variational formulation that can be seen as a generalisation of the Laplacian Navier-Stokes form to fluids with shear-rate-dependent viscosity, as appearing in hemodynamic and polymeric flows. By appropriately manipulating the viscous terms in the variational formulation and employing a simple projection of the constitutive law, it is possible to devise a formulation with the desired natural boundary conditions and low computational complexity. Several numerical examples are presented to showcase the potential of our method, revealing improved accuracy and robustness in comparison with the state of the art.

A Mathematical Model to Simulate Static Characteristics of T-Beam Bridge with Wide Flange

This study considers various factors, such as shear lag effect and shear deformation, and introduces the self-stress equilibrium for shear lag warping stress conditions to analyze the static characteristics of T-beam bridges accurately. In the mechanical analysis, three generalized displacement functions are applied, and the governing differential equations and natural boundary conditions of the static characteristics of T-beams are established on the basis of the energy variational principle. In the example, the influences of the shear lag effect, different load forms, and span ratio on the mechanical properties of T-beam bridges are analyzed. Therefore, the method of this study enriches and develops the theoretical analysis of T-beams, and it plays a certain guiding role in designing such a structure.

A non-standard class of variational problems of Herglotz type

<p style='text-indent:20px;'>In this paper, we extend the variational problem of Herglotz considering the case where the Lagrangian depends not only on the independent variable, an unknown function <inline-formula><tex-math id="M1">\begin{document}$ x $\end{document}</tex-math></inline-formula> and its derivative and an unknown functional <inline-formula><tex-math id="M2">\begin{document}$ z $\end{document}</tex-math></inline-formula>, but also on the end points conditions and a real parameter. Herglotz's problems of calculus of variations of this type cannot be solved using the standard theory. Main results of this paper are necessary optimality condition of Euler-Lagrange type, natural boundary conditions and the Dubois-Reymond condition for our non-standard variational problem of Herglotz type. We also prove a necessary optimality condition that arises as a consequence of the Lagrangian dependence of the parameter. Our results not only provide a generalization to previous results, but also give some other interesting optimality conditions as special cases. In addition, two examples are given in order to illustrate our results.</p>

Elements of Bi-cubic Polynomial Natural Spline Interpolation for Scattered Data: Boundary Conditions Meet Partition of Unity Technique

This paper investigates one kind of interpolation for scattered data by bi-cubic polynomial natural spline, in which the integral of square of partial derivative of two orders to x and to y for the interpolating function is minimal (with natural boundary conditions). Firstly, bi-cubic polynomial natural spline interpolations with four kinds of boundary conditions are studied. By the spline function methods of Hilbert space, their solutions are constructed as the sum of bi-linear polynomials and piecewise bi-cubic polynomials. Some properties of the solutions are also studied. In fact, bi-cubic natural spline interpolation on a rectangular domain is a generalization of the cubic natural spline interpolation on an interval. Secondly, based on bi-cubic polynomial natural spline interpolations of four kinds of boundary conditions, and using partition of unity technique, a Partition of Unity Interpolation Element Method (PUIEM) for fitting scattered data is proposed. Numerical experiments show that the PUIEM is adaptive and outperforms state-of-the-art competitions, such as the thin plate spline interpolation and the bi-cubic polynomial natural spline interpolations for scattered data.

Finite difference approach to fourth-order linear boundary-value problems

Discrete approximations to the equation $$\begin{equation*}L_\textrm{cont}u = u^{(4)} + D(x) u^{(3)} + A(x) u^{(2)} + (A^{\prime}(x)+H(x)) u^{(1)} + B(x) u = f, \quad x\in[0,1]\end{equation*}$$are considered. This is an extension of the Sturm–Liouville case $D(x)\equiv H(x)\equiv 0$ (Ben-Artzi et al. (2018) Discrete fourth-order Sturm–Liouville problems. IMA J. Numer. Anal., 38, 1485–1522) to the non-self-adjoint setting. The ‘natural’ boundary conditions in the Sturm–Liouville case are the values of the function and its derivative. The inclusion of a third-order discrete derivative entails a revision of the underlying discrete functional calculus. This revision forces evaluations of accurate discrete approximations to the boundary values of the second-, third- and fourth-order derivatives. The resulting functional calculus provides the discrete analogs of the fundamental Sobolev properties of compactness and coercivity. It allows us to obtain a general convergence theorem of the discrete approximations to the exact solution. Some representative numerical examples are presented.

A Thermodynamics-Based Nonlocal Bar-Elastic Substrate Model with Inclusion of Surface-Energy Effect

This paper presents a bar-elastic substrate model to investigate the axial responses of nanowire-elastic substrate systems considering the effects of nonlocality and surface energy. The thermodynamics-based strain gradient model is adopted to capture the nonlocality of the bar-bulk material while the Gurtin-Murdoch surface theory is utilized to consider the surface energy. To characterize the bar-surrounding substrate interaction, the Winkler foundation model is employed. In a direct manner, system compatibility conditions are obtained while within the framework of the virtual displacement principle, the system equilibrium condition and the corresponding natural boundary conditions are consistently obtained. Three numerical simulations are conducted to investigate the characteristics and behaviors of the nanowire-elastic substrate system: the first is conducted to reveal the capability of the proposed model to eliminate the paradoxical behavior inherent to the Eringen nonlocal differential model; the second is employed to characterize responses of the nanowire-elastic substrate system; and the third is aimed at demonstrating the dependence of the system effective Young’s modulus on several system parameters.

Management of the Design Parameters in Optimal Design Problems

The solution to the problem of designing rational load-bearing structures should be associated with the direct use of the principles that govern the deformation of a solid. If the functional of the direct problem has as Euler – Lagrange equations and natural boundary conditions the equations and boundary conditions of the accepted deformation theory, then they must correspond to the functional of the design problem, in addition, to additional equations indicating the dependence of the system energy change on the configuration change and the elastic modules of the body material. Possible variations of the configuration functions and modules of elasticity of the material will be infinitely small changes of the functions satisfying the prescriptive requirements to the structure and material; they are continuous and satisfy the requirements of differentiability. Due to the small variations in the functions that determine the configuration, we neglect changes in the arrangement of external forces relative to individual parts of the body and changes in the temperature field.