Search Results (28)

This study investigates three-dimensional two-phase flows in complex geometries found in industrial process equipment design using finite-element numerical simulations. The governing equations are formulated in three-dimensional Cartesian coordinates and solved on unstructured meshes employing the Taylor–Hood “Mini” element, selected for its numerical stability and convergence properties. The convective term in the momentum equation is discretized using a first-order semi-Lagrangian scheme. The two fluid phases are separated by an interface mesh composed of triangular surface elements, which is independent of the primary volumetric fluid mesh. Surface tension effects are incorporated as a source term using the continuum surface force (CSF) model, with the curvature computed via the Laplace–Beltrami operator. At each time step, the positions of the interface mesh nodes are updated according to the local fluid velocity field. The results show that the methodology is stable and can be used to accurately model two-phase flows in complex geometries found in several engineering solutions. Full article

Cell membranes contain a variety of biomolecules, especially various kinds of lipids and proteins, which constantly change with fluidity and environmental stimuli. Though Helfrich curvature elastic energy has successfully explained many phenomena for single-component membranes, a new theoretical framework for multicomponent membranes is still a challenge. In this work, we propose a generalized Helfrich free-energy functional describe equilibrium shapes and phase behaviors related to membrane heterogeneity with via curvature-component coupling in a unified framework. For multicomponent membranes, a new but important Laplace–Beltrami operator is derived from the variational calculation on the integral of Gaussian curvature and applied to explain the spontaneous nanotube formation of an asymmetric glycolipid vesicle. Therefore, our general mathematical framework shows a predictive capabilities beyond the existing multicomponent membrane models. The set of new curvature-component coupling EL equations have been derived for global vesicle shapes associated with the composition redistribution of multicomponent membranes for the first time and specified into several typical geometric shape equations. The equilibrium radii of isotonic vesicles for both spherical and cylindrical geometries are calculated. The analytical solution for isotonic vesicles reveals that membrane stability requires distinct elastic moduli among components ($k_{\mathrm{A}}\ne k_{\mathrm{B}}$, ${\overline{k}}_{\mathrm{A}}\ne {\overline{k}}_{\mathrm{B}}$), which is consistent with experimental observations of coexisting lipid domains. Furthermore, we elucidate the biophysical implications of the derived shape equations, linking them to experimentally observed membrane remodeling processes. Our new free-energy framework provides a baseline for more detailed microscopic membrane models. Full article

In this paper, we investigate a perturbed parabolic problem involving the Laplace–Beltrami operator on a smooth compact Riemannian manifold M. For a strongly local Dirichlet form in $L^2\left(M\right)$. More precisely, we begin by proving that, in the case of the existence of a non-negative solution, the potential can be written as a derivative of some functions which are locally integrable on M; after that, we prove the existence of a non-negative solution for such problems. Full article

This study conducts an in-depth analysis of clustering small molecules using spectral geometry and deep learning techniques. We applied a spectral geometric approach to convert molecular structures into triangulated meshes and used the Laplace–Beltrami operator to derive significant geometric features. By examining the eigenvectors of these operators, we captured the intrinsic geometric properties of the molecules, aiding their classification and clustering. The research utilized four deep learning methods: Deep Belief Network, Convolutional Autoencoder, Variational Autoencoder, and Adversarial Autoencoder, each paired with k-means clustering at different cluster sizes. Clustering quality was evaluated using the Calinski–Harabasz and Davies–Bouldin indices, Silhouette Score, and standard deviation. Nonparametric tests were used to assess the impact of topological descriptors on clustering outcomes. Our results show that the DBN + k-means combination is the most effective, particularly at lower cluster counts, demonstrating significant sensitivity to structural variations. This study highlights the potential of integrating spectral geometry with deep learning for precise and efficient molecular clustering. Full article

In the four-dimensional Minkowski space, hypersurfaces classified as right conoids with a time-like axis are introduced and studied. The computation of matrices associated with the fundamental form, the Gauss map, and the shape operator specific to these hypersurfaces is included in our analysis. The intrinsic curvatures of these hypersurfaces are determined to provide a deeper understanding of their geometric properties. Additionally, the conditions required for these hypersurfaces to be minimal are established, and detailed calculations of the Laplace–Beltrami operator are performed. Illustrative examples are provided to enhance our comprehension of these concepts. Finally, the umbilical condition is examined to determine when these hypersurfaces become umbilic, and also the Willmore functional is explored. Full article

This work’s goal is to numerically investigate the interactions between two gas bubbles in a fluid flow in a circular cross-section channel, both in the presence and in the absence of gravitational forces, with several Reynolds and Weber numbers. The first bubble is placed at the center of the channel, while the second is near the wall. Their positions are set in such a way that a dynamic interaction is expected to occur due to their velocity differences. A finite element numerical tool is utilized to solve the incompressible Navier–Stokes equations and simulate two-phase flow using an unfitted mesh to represent the fluid interface, akin to the front-tracking method. The results show that the velocity gradient influences bubble shapes near the wall. Moreover, lower viscosity and surface tension force account for more significant interactions, both in the bubble shape and in the trajectory. In this scenario, it can be observed that one bubble is trapped in the other’s wake, with the proximity possibly allowing the onset of coalescence. The results obtained contribute to a deeper understanding of two-phase inner flows. Full article

The twisted hypersurfaces $\mathfrak{x}$ with the $(0,0,0,0,1)$ rotating axis in five-dimensional Euclidean space ${\mathbb{E}}^5$ is considered. The fundamental forms, the Gauss map, and the shape operator of $\mathfrak{x}$ are calculated. In ${\mathbb{E}}^5$, describing the curvatures by using the Cayley–Hamilton theorem, the curvatures of hypersurfaces $\mathfrak{x}$ are obtained. The solutions of differential equations of the curvatures of the hypersurfaces are open problems. The umbilically and minimality conditions to the curvatures of $\mathfrak{x}$ are determined. Additionally, the Laplace–Beltrami operator relation of $\mathfrak{x}$ is given. Full article

Insert gradient coils with similar imaging body shapes typically have smaller dimensions and higher spatial efficiency. This often allows the gradient coils the achievement of stronger and faster gradient fields. Thus, improving existing methods to make them applicable to the design of MRI gradient coils on complex surfaces has also become a challenge. This article proposes an algorithm that smooths the implicitly expressed stream function based on the intrinsic surface Laplace–Beltrami operator. This algorithm can be used to simplify the design procedure of MRI gradient coils on non-developable surfaces. The following steps are performed by the proposed algorithm: an initial design of the stream function configuration, extraction of the surface mesh, discretization of the surface smoothing operator, and a smoothing of the contour lines. To evaluate the quality of the smoothed streamline configuration, several technical parameter metrics—including magnetic field accuracy, coil power consumption, theoretical minimum wire spacing, and the maximum curvature of the contour lines—were evaluated. The proposed method was successfully validated in a design gradient coil on both developable and non-developable surfaces. All examples evolved from an initial value with a locally non-smooth and complex topological configuration to a smooth result while maintaining high magnetic field accuracy. Full article

We present a family of hypersurfaces of revolution distinguished by four parameters in the five-dimensional pseudo-Euclidean space ${\mathbb{E}}_2^5$. The matrices corresponding to the fundamental form, Gauss map, and shape operator of this family are computed. By utilizing the Cayley–Hamilton theorem, we determine the curvatures of the specific family. Furthermore, we establish the criteria for maximality within this framework. Additionally, we reveal the relationship between the Laplace–Beltrami operator of the family and a $5\times 5$ matrix. Full article

We investigate the class of helical hypersurfaces parametrized by $\mathfrak{x}=\mathfrak{x}(u,v,w)$, characterized by a light-like axis in Minkowski spacetime ${\mathbb{L}}^4$. We determine the matrices that represent the fundamental forms, Gauss map, and shape operator of $\mathfrak{x}$. Furthermore, employing the Cayley–Hamilton theorem, we compute the curvatures associated with $\mathfrak{x}$. We explore the conditions under which the curvatures of $\mathfrak{x}$ possess the property of being umbilical. Moreover, we provide the Laplace–Beltrami operator for the family of helical hypersurfaces with a light-like axis in ${\mathbb{L}}^4$. Full article

The method of gradient estimation for the heat-type equation using the Harnack quantity is a classical approach used for understanding the nature of the solution of these heat-type equations. Most of the studies in this field involve the Laplace–Beltrami operator, but in our case, we studied the weighted heat equation that involves weighted Laplacian. This produces a number of terms involving the weight function. Thus, in this article, we derive the Harnack estimate for a positive solution of a weighted nonlinear parabolic heat equation on a weighted Riemannian manifold evolving under a geometric flow. Applying this estimation, we derive the Li–Yau-type gradient estimation and Harnack-type inequality for the positive solution. A monotonicity formula for the entropy functional regarding the estimation is derived. We specify our results for various different flows. Our results generalize some works. Full article

We introduce the generalized helical hypersurface having a space-like axis in five-dimensional Minkowski space. We compute the first and second fundamental form matrices, Gauss map, and shape operator matrix of the hypersurface. Additionally, we compute the curvatures of the hypersurface by using the Cayley–Hamilton theorem. Moreover, we give some relations for the mean and the Gauss–Kronecker curvatures of the hypersurface. Finally, we obtain the Laplace–Beltrami operator of the hypersurface. Full article

We consider the problem of showing that 1 is an eigenvalue for a family of generalised transfer operators of the Farey map. This is an important problem in the thermodynamic formalism approach to dynamical systems, which in this particular case is related to the spectral theory of the modular surface via the Selberg Zeta function and the theory of dynamical zeta functions of maps. After briefly recalling these connections, we show that the problem can be formulated for operators on an appropriate Hilbert space and translated into a linear algebra problem for infinite matrices. This formulation gives a new way to study numerically the spectrum of the Laplace–Beltrami operator and the properties of the Selberg Zeta function for the modular surface. Full article

In this paper, the generalized helical hypersurfaces $\mathbf{x}=\mathbf{x}(u,v,w)$ with a time-like axis in Minkowski spacetime ${\mathbb{E}}_1^4$ are considered. The first and the second fundamental form matrices, the Gauss map, and the shape operator matrix of $\mathbf{x}$ are calculated. Moreover, the curvatures of the generalized helical hypersurface $\mathbf{x}$ are obtained by using the Cayley–Hamilton theorem. The umbilical conditions for the curvatures of $\mathbf{x}$ are given. Finally, the Laplace–Beltrami operator of the generalized helical hypersurface with a time-like axis is presented in ${\mathbb{E}}_1^4$. Full article

The computation of correspondences between shapes is a principal task in shape analysis. In this work, we consider correspondences constructed by a numerical solution of partial differential equations (PDEs). The underlying model of interest is thereby the classic wave equation, since this may give the most accurate shape matching. As has been observed in previous works, numerical time discretisation has a substantial influence on matching quality. Therefore, it is of interest to understand the underlying mechanisms and to investigate at the same time if there is an analytical model that could best describe the most suitable method for shape matching. To this end, we study here the damped wave equation, which mainly serves as a tool to understand and model properties of time discretisation. At the hand of a detailed study of possible parameters, we illustrate that the method that gives the most reasonable feature descriptors benefits from a damping mechanism which can be introduced numerically or within the PDE. This sheds light on some basic mechanisms of underlying computational and analytic models, as one may conjecture by our investigation that an ideal model could be composed of a transport mechanism and a diffusive component that helps to counter grid effects. Full article