Concave Domains Research Articles

GEOMETRIC CONTROL OF BONE TISSUE GROWTH AND ORGANIZATION

Cells typically respond to a variety of geometrical cues in their environment, ranging from nanoscale surface topography to mesoscale surface curvature. The ability to control cellular organisation and fate by engineering the shape of the extracellular milieu offers exciting opportunities within tissue engineering. Despite great progress, however, many questions regarding geometry-driven tissue growth remain unanswered.Here, we combine mathematical surface design, high-resolution microfabrication, in vitro cell culture, and image-based characterization to study spatiotemporal cell patterning and bone tissue formation in geometrically complex environments. Using concepts from differential geometry, we rationally designed a library of complex mesostructured substrates (101-103 µm). These substrates were accurately fabricated using a combination of two-photon polymerisation and replica moulding, followed by surface functionalisation. Subsequently, different cell types (preosteoblasts, fibroblasts, mesenchymal stromal cells) were cultured on the substrates for varying times and under varying osteogenic conditions. Using imaging-based methods, such as fluorescent confocal microscopy and second harmonic generation imaging, as well as quantitative image processing, we were able to study early-stage spatiotemporal cell patterning and late-stage extracellular matrix organisation. Our results demonstrate clear geometry-dependent cell patterning, with cells generally avoiding convex regions in favour of concave domains. Moreover, the formation of multicellular bridges and collective curvature-dependent cell orientation could be observed. At longer time points, we found clear and robust geometry-driven orientation of the collagenous extracellular matrix, which became apparent with second harmonic generation imaging after ∼2 weeks of culture.Our results highlight a key role for geometry as a cue to guide spatiotemporal cell and tissue organisation, which is relevant for scaffold design in tissue engineering applications. Our ongoing work aims at understanding the underlying principles of geometry-driven tissue growth, with a focus on the interactions between substrate geometry and mechanical forces.

Experimental investigation on the interaction between mode-2 internal solitary wave and horizontal transverse cylinder

The experimental study on the interaction between a mode-2 internal solitary wave (ISW) and a horizontal transverse cylinder has been carried out in a large gravity-stratified fluid flume. One kind of making-wave method of “rotating-blade-gate” is proposed to achieve the experimental simulation of the standard mode-2 ISW in the laboratory flume. The conductivity probe array and particle image velocimetry are used to measure the mode-2 ISW and its wave-flow structure of the interaction with the horizontal transverse cylinder, and the micro-amplitude force sensor is used to measure the forces of the mode-2 ISW on the cylinder model. It is shown that the mode-2 ISW geometric structure is characterized by a convex and concave oval wave envelope shape and the counterclockwise and clockwise circulations are formed in its convex and concave domains, respectively. The resulting pair of upper and lower antisymmetric circulations moves forward together, which causes a horizontal strong flow at the symmetric center of the wave envelope consistent with the wave propagation direction. The horizontal transverse cylinder is subjected to the combined action of horizontal flow, vertical flow, and density change induced by the mode-2 ISW, in which the horizontal strong flow, induced flow separation, and vortex structure located at the symmetric center of the wave envelope are the signature features that are different from other positions. The theoretical analysis model of the forces exerted by the mode-2 ISW on a horizontal transverse cylinder is established. The spatial distribution characteristic of the force exerted by the mode-2 ISW on a horizontal cylinder is obtained, and the essential reason for the maximum horizontal and vertical loads on the cylinder is revealed. The horizontal maximum load corresponds to the inhomogeneous change of the horizontal velocity along the depth, and the vertical maximum load depends on the maximum density change along the depth, of which variation trend corresponds to the linear change of wave amplitudes.

Solution of Exterior Quasilinear Problems Using Elliptical Arc Artificial Boundary

In this paper, the method of artificial boundary conditions for exterior quasilinear problems in concave angle domains is investigated. Based on the Kirchhoff transformation, the exact quasiliner elliptical arc artificial boundary condition is derived. Using the approximate elliptical arc artificial boundary condition, the finite element method is formulated in a bounded region. The error estimates are obtained. The effectiveness of our method is showed by some numerical experiments.

A generic strategy to obtain semi‐analytical mesh sensitivities/velocities for tetrahedral mesh generators

AbstractThis article proposes a generic approach to obtain semi‐analytical mesh sensitivities and shows how to compute mesh updates for any tetrahedral mesher. The proposed strategy allows sensitivities to be computed for a given mesh or first improve an initial mesh and then compute sensitivities. The strategy is based on a modification of Persson and Strang's mesher, Distmesh, analogizing the mesh to a simple truss structure. The boundary of the mesh is maintained using Lagrangian multipoint constraints. An analytical relationship between the mesh and the boundary permits the nodes to move freely along the boundary. This links the mesh directly to the parameterized geometry. We demonstrate the approach's ability to produce accurate, smooth and continuous mesh sensitivities and update the mesh. We use the resulting sensitivities to predict mesh updates corresponding to geometry updates. Four examples are given that show the method can accurately describe linear and nonlinear convex and concave domains. An example is given that also demonstrates the ability to generate accurate mesh sensitivities for a mesh created using the third party mesher GMSH.

A modification of the method of fundamental solutions for solving 2D problems with concave and complicated domains

To solve various problems in complicated and concave domains by the method of fundamental solutions (MFS), it is required to consider a large number of source and collocation points that increases the computational time of the analysis. This paper suggests a modification to the MFS, which can make it more efficient and reliable for solving applied problems in complicated domains. In the proposed method, each pseudo source is converted to two or more number of sub-sources with equal intensities. Therefore, the total number of sources is increased while the number of unknowns is not increased. By this approach we will be able to reduce the distance between the main and the pseudo boundaries and therefore modeling the problems with complicated boundaries can be performed effectively. The proposed method is investigated for a scalar field problem, i.e. the Laplace equation, and a vector field problem, i.e. elastostatic problem. It is shown that by using the proposed method, in addition to reducing the calculation time, the condition number of the coefficient matrix also significantly decreases. By solving several numerical example problems, it is observed that the presented method can obtain accurate solutions by using a small number of collocation points.

Thermal postbuckling analysis of FG-CNTRC plates with various shapes and temperature-dependent properties using the VDQ-FEM technique

Using the ideas of variational differential quadrature and finite element methods (VDQ and FEM), a novel numerical approach is developed in this paper to investigate the postbuckling behavior of plates with arbitrary shape under the action of thermal load within the framework of higher-order shear deformation theory (HSDT). It is considered that the plates are made of functionally graded carbon nanotube reinforced composite (FG-CNTRC). By the proposed approach, the thermal postbuckling behavior of plates with arbitrary-shaped cutout can be modeled. The rule of mixture is utilized to obtain the effective material properties of nanocomposite which are considered to be temperature-dependent. To implement the proposed method that can be named as VDQFEM, the variational temperature-dependent formulation of problem is first developed. This formulation is presented using novel vector-matrix relations with the aim of utilizing in programming in an efficient way. In the context of VDQFEM, the space domain of plate is first transformed into a number of finite elements. In the next step, the VDQ discretization technique is implemented within each element. Then, the assemblage procedure is performed to derive the set of matricized governing equations which is finally solved by means of the pseudo arc-length continuation algorithm. One of the main novelties of present approach is proposing an efficient way based on mixed-formulation to accommodate the continuity of first-order derivatives on the common boundaries of elements for the used higher-order shear deformable plate model. Quadrilateral plate, skew plate, annular plate, square plate with rectangular hole and rectangular plate with circular hole under different boundary conditions including CCCC, SSSS, CSCS and CCSS (S: simply-supported, C: clamped) are considered in the numerical example. First, validation studies are presented in terms of critical buckling temperature of CNTRC rectangular plates with various CNT distribution patterns. It is shown that the proposed method has the advantages of standard VDQ technique including fast convergence rate, being locking-free and simple implementation. Moreover, it is capable of considering polygon and concave domains. Also, the effects of geometrical properties, volume fraction and distribution pattern of CNTs and temperature-dependency of material properties on the thermal postbuckling of plates are studied.

Traveling and standing fronts on curved surfaces

We analyze heteroclinic traveling waves propagating on two dimensional manifolds to show that the geometric modification of the front velocity is proportional to the geodesic curvature of the front line. As a result, on surfaces of concave domains, stable standing fronts can be formed on lines of constant geodesic curvature. These lines minimize the geometric functional describing the system’s energy, consisting of terms proportional to the front line-length and to the inclosed surface area. Front stabilization at portions of surface with negative Gaussian curvature, provides a mechanism of pattern formation. In contrast to the mechanism associated with the Turing instability, the proposed mechanism requires only a single scalar bistable reaction–diffusion equation and connects the intrinsic surface geometry with the arising pattern. By considering a system of equations modeling boundary-volume interactions, we show that polarization of the boundary may induce a corresponding polarization in the volume.

Polarization of concave domains by traveling wave pinning.

Pattern formation is one of the most fundamental yet puzzling phenomena in physics and biology. We propose that traveling front pinning into concave portions of the boundary of 3-dimensional domains can serve as a generic gradient-maintaining mechanism. Such a mechanism of domain polarization arises even for scalar bistable reaction-diffusion equations, and, depending on geometry, a number of stationary fronts may be formed leading to complex spatial patterns. The main advantage of the pinning mechanism, with respect to the Turing bifurcation, is that it allows for maintaining gradients in the specific regions of the domain. By linking the instant domain shape with the spatial pattern, the mechanism can be responsible for cellular polarization and differentiation.

Stability analysis of the thin plates with arbitrary shapes subjected to non-uniform stress fields using boundary element and radial integration methods

In this paper, a boundary element method is applied to buckling analysis of thin plates with arbitrary shapes under various load types. The governing differential equation is converted into equivalent integral equation using the static fundamental solutions of the biharmonic equation. The arising domain integrals due to in-plane stresses are evaluated applying the radial integration method. The in-plane stresses are implemented through Gaussian integration points along radial direction in the convex domains. For the concave domains, an idea has been introduced to compute radial integration along new direction from an auxiliary point to the field point. This method can be easily applied to buckling analysis of thin plates with arbitrarily distributed and concentrated edge loading. Six sample problems are presented to illustrate the effectiveness and accuracy of the proposed method.

Role of thermal and flow characteristics on entropy generation during natural convection in porous enclosures with curved walls subjected to Rayleigh-Bénard heating

The comprehensive analysis of natural convection within various enclosures with the concave and convex horizontal walls involving the classical Rayleigh-Bénard heating (hot bottom wall, cold top wall and adiabatic side walls) is carried out via the entropy generation approach. Galerkin finite element method is employed to obtain the solutions in terms of the isotherms (θ), streamlines (ψ), entropy generation (Sθ,Sψ and Stotal) and average heat transfer rates (Nut‾) for a selected range of Darcy numbers (10-5⩽Dam⩽10-2), Prandtl numbers (Prm=0.015 and 7.2) at Rayleigh number, Ram=106 with various test cases with different wall curvatures (concave and convex). The magnitudes and spatial locations of local entropy generation due to heat transfer and fluid friction are studied for various thermal and geometrical parameters. The active zones with larger Sθ are found along near core region and middle portions of the curved walls with less and moderate concavities whereas, those are found only near the middle portions of the curved walls with the high concavity at the low Dam involving all Prm. In the convex domain, the active zones with larger Sθ are seen near the corner regions for the low Dam at all Prm. The zones prone to the entropy generation with the high Sθ are seen along the mid-horizontal axis of the cavity involving all the concave and convex domains at the high Dam and all Prm. Also, the left and right portions of the curved walls act as the active zones with the high Sθ for the concave domains at the high Dam and all Prm. The larger values of Sψ are seen near all the solid walls (at all Dam) and at the interior zone (at the high Dam) for all Prm involving all concave and convex cases. Based on the comparison of all the concave and convex cases, the concave case with the high wall concavity may be preferred over the specific concave cases (with the less and moderate wall concavities) and all the convex cases due to the larger Nut‾ and less Stotal values for the entire range of Dam and Prm.

Strichartz and localized energy estimates for the wave equation in strictly concave domains

We prove localized energy estimates for the wave equation in domains with a strictly concave boundary when homogeneous Dirichlet or Neumann conditions are imposed. By restricting the solution to small, frequency dependent, space time collars of the boundary, it is seen that a stronger gain in regularity can be obtained relative to the usual energy estimates. Mixed norm estimates of Strichartz and square function type follow as a result, using the energy estimates to control error terms which arise in a wave packet parametrix construction. While the latter estimates are not new for Dirichlet conditions, the present approach provides an avenue for treating these estimates when Neumann conditions are imposed. The method also treats Schr{\"o}dinger equations with time independent coefficients.

Proper holomorphic mappings into $\\ell$-concave quadric domains in projective space

In this paper, we prove a type of partial rigidity result for proper holomorphic mappings of certain $\ell$-concave domains in projective space into model quadratic $\ell$-concave domains. The main technical result is a degree estimate for proper holomorphic mappings into the model domains, provided that the mappings extend to projective space as rational mappings, and the source domain contains algebraic varieties and has a boundary with low CR complexity.

Haar Wavelet for the Natural Boundary Integral Equation

In this paper, we construct Haar wavelet and apply it to investigate the numerical solution of the natural boundary integral equation of the Laplace equation in the concave angle domains. Haar wavelet has better stability and good explicit expression. Moreover, they are mutual orthogonal. We make full use of their mutual orthogonal to cope with the natural boundary integral equation. Taking advantage of Galerkin-wavelet method in discretizing the natural boundary integral equation. Finally, a numerical example is shown and the feasibility and validity of the method are proved.

GRAND — Ground structure based topology optimization for arbitrary 2D domains using MATLAB

The present work describes in detail an implementation of the ground structure method for non---orthogonal unstructured and concave domains written in MATLAB, called GRAND -- GRound structure ANaly...

Incorporating improved refinement techniques for a grid-based geometrically-adaptive hexahedral mesh generation algorithm

This paper presents a density control based adaptive hexahedral mesh generation algorithm for three dimensional models. To improve the mesh quality of hexahedral elements, a set of improved 27-refinement templates is proposed and the refinement modes of these templates are given. A set of effective refinement templates for 8-refinement based mesh generation algorithm is employed. The corresponding date structure and the procedures for realization of the algorithm are also presented. A buffer layer is inserted on the concave domains to resolve the propagation problems. Finally, the effectiveness and robustness of the algorithm are demonstrated by using several examples.

On nodal line of the second eigenfunction of the Laplacian over concave domains in ℝ2

In this note, the Payne’s conjecture from convex domains to concave domains is generalized. It is shown that the nodal line N of the second eigenfunction of the Laplacian over some simply connected concave domain Ω in ℝ2 must intersect the boundary ∂Ω at exactly two points.

Nonconforming finite element approximations of the Steklov eigenvalue problem and its lower bound approximations

The paper deals with error estimates and lower bound approximations of the Steklov eigenvalue problems on convex or concave domains by nonconforming finite element methods. We consider four types of nonconforming finite elements: Crouzeix-Raviart, Q 1 rot , EQ 1 rot and enriched Crouzeix-Raviart. We first derive error estimates for the nonconforming finite element approximations of the Steklov eigenvalue problem and then give the analysis of lower bound approximations. Some numerical results are presented to validate our theoretical results.

Some results of analytic functions related to concave domains

The main purpose of the present paper is to derive some results for analytic functions whose takes values on concave domains. As special cases of these results, several new sufficient conditions for univalency and starlikeness of analytic functions are obtained.

Estimates for regularity of the tangential \documentclass{article}\usepackage{amssymb,amsmath}\begin{document}\pagestyle{empty}$\bar{\partial }$\end{document}‐system

AbstractWe study ellipticity in a weak sense, such as fractional or logarithmic, of the system \documentclass{article}\usepackage{amssymb,amsmath}\begin{document}\pagestyle{empty}$\big (\bar{\partial }_b,\bar{\partial }^*_b\big )$\end{document} tangential to a hypersurface or a generic higher codimensional submanifold \documentclass{article}\usepackage{amssymb,amsmath}\begin{document}\pagestyle{empty}$M\subset {\mathbb {C}}^n$\end{document}. The geometric setting which assures the estimates is the q‐pseudoconvexity/concavity of M in addition to the existence of a suitable family of weights in a strip or a tube around M. The basic estimates for the \documentclass{article}\usepackage{amssymb,amsmath}\begin{document}\pagestyle{empty}$\bar{\partial }$\end{document}‐Neumann problem on q‐pseudoconvex/concave domains is related to the classical work by Shaw 17 and more recent by Zampieri 19. The method of the weights is due to Catlin 3 and the relation between the tangential and the ambient \documentclass{article}\usepackage{amssymb,amsmath}\begin{document}\pagestyle{empty}$\bar{\partial }$\end{document} system on pseudoconvex domains is inspired to Kohn 14. Both these techniques are adapted here to a general Levi signature.

Investigation of the stress‐strain state of the laminated shallow shells by R‐functions method combined with spline‐approximation

AbstractThe bending behavior of the laminated shallow shells under static loading has been studied using the R‐functions theory together with the spline‐approximation. Formulation is based on the first order shear deformation theory. Due to usage of the R‐functions theory the laminated shallow shells with complex shape and different types of the boundary conditions can be investigated. Application of the spline‐approximation allows getting reliable and validated results for non concave domains and domains with holes. The proposed method is implemented in the appropriate software in framework of the mathematical package MAPLE. The analysis of influence of certain factors (curvature, packing of layers, geometrical parameters, boundary conditions) on a stress‐strain state is carried out for shallow shells with cut‐outs. The comparison of obtained results with those already known from literature and results obtained by using ANSYS are also presented.