Showing papers on "Boundary value problem published in 2017"

Application of reproducing kernel algorithm for solving second-order, two-point fuzzy boundary value problems

Abstract: In this paper, we investigate the analytic and approximate solutions of second-order, two-point fuzzy boundary value problems based on the reproducing kernel theory under the assumption of strongly generalized differentiability. The solution methodology is based on generating the orthogonal basis from the obtained kernel functions, while the orthonormal basis is constructing in order to formulate and utilize the solutions with series form in terms of their r-cut representation in the space $$\oplus _{j=1}^2 W_2^3 \left[ {a,b}\right] $$źj=12W23a,b. An efficient computational algorithm is provided to guarantee the procedure and to confirm the performance of the proposed method. Results of numerical experiments are provided to illustrate the theoretical statements in order to show potentiality, generality, and superiority of our algorithm for solving such fuzzy equations. Graphical results, tabulated data, and numerical comparisons are presented and discussed quantitatively to illustrate the possible fuzzy solutions.

Control Design for Nonlinear Flexible Wings of a Robotic Aircraft

Abstract: In this brief, the control problem for flexible wings of a robotic aircraft is addressed by using boundary control schemes Inspired by birds and bats, the wing with flexibility and articulation is modeled as a distributed parameter system described by hybrid partial differential equations and ordinary differential equations Boundary control for both wing twist and bending is proposed on the original coupled dynamics, and bounded stability is proved by introducing a proper Lyapunov function The effectiveness of the proposed control is verified by simulations

Stress-driven versus strain-driven nonlocal integral model for elastic nano-beams

Abstract: In the strain-driven model of nonlocal elasticity proposed by Eringen , the elastic strain is defined by a Fredholm integral equation in which the stress is the output of a convolution between the local response to an elastic strain and a smoothing kernel dependent on a nonlocal parameter. In the wake of this proposal, size effects in nano-beams were investigated in literature by adopting a differential formulation considered to be equivalent to the integral one. Recent improvements have however revealed that equivalence requires also the fulfilment of constitutive boundary conditions. Moreover, this strain-driven nonlocal elastic problem has been shown to be ill-posed, being conflicting with equilibrium requirements. A stress-driven integral constitutive law provides the natural way to get well-posed nonlocal elastic problems for application to nano-structures. The new integral constitutive law is formulated with explicit reference to plane and straight nano-beams according to the standard Bernoulli - Euler structural model. The solution procedure based on the stress-driven nonlocal law is described and adopted for the solution of a simple statically indeterminate scheme, thus showing effectiveness of the new model for the structural design of nano-devices.

Bulk locality from modular flow

Abstract: We study the reconstruction of bulk operators in the entanglement wedge in terms of low energy operators localized in the respective boundary region. To leading order in N, the dual boundary operators are constructed from the modular flow of single trace operators in the boundary subregion. The appearance of modular evolved boundary operators can be understood due to the equality between bulk and boundary modular flows and explicit formulas for bulk operators can be found with a complete understanding of the action of bulk modular flow, a difficult but in principle solvable task. We also obtain an expression when the bulk operator is located on the Ryu-Takayanagi surface which only depends on the bulk to boundary correlator and does not require the explicit use of bulk modular flow. This expression generalizes the geodesic operator/OPE block dictionary to general states and boundary regions.

Boundary Conditions: What They Are, How to Explore Them, Why We Need Them, and When to Consider Them

Abstract: Boundary conditions (BC) have long been discussed as an important element in theory development, referring to the “who, where, when” aspects of a theory. However, it still remains somewhat vague as...

Topology optimization of multi-material negative Poissons ratio metamaterials using a reconciled level set method

Abstract: Metamaterials are defined as a family of rationally designed artificial materials which can provide extraordinary effective properties compared with their nature counterparts. This paper proposes a level set based method for topology optimization of both single and multiple-material Negative Poissons Ratio (NPR) metamaterials. For multi-material topology optimization, the conventional level set method is advanced with a new approach exploiting the reconciled level set (RLS) method. The proposed method simplifies the multi-material topology optimization by evolving each individual material with a single level set function and reconciling the result level set field with the MerrimanBenceOsher (MBO) operator. The NPR metamaterial design problem is recast as a variational problem, where the effective elastic properties of the spatially periodic microstructure are formulated as the strain energy functionals under uniform displacement boundary conditions. The adjoint variable method is utilized to derive the shape sensitivities by combining the general linear elastic equation with a weak imposition of Dirichlet boundary conditions. The design velocity field is constructed using the steepest descent method and integrated with the level set method. Both single and multiple-material mechanical metamaterials are achieved in 2D and 3D with different Poissons ratios and volumes. Benchmark designs are fabricated with multi-material 3D printing at high resolution. The effective auxetic properties of the achieved designs are verified through finite element simulations and characterized using experimental tests as well. A multi-material topology optimization approach exploiting the reconciled level-set method.The boundary of each individual material is evolved with a single level set function.Multiple level set functions are reconciled with the MerrimanBenceOsher (MBO) operator.Both 2D and 3D multi-material designs were obtained and used for validate the proposed method.

Why does bulk boundary correspondence fail in some non-hermitian topological models

Abstract: Bulk boundary correspondence is crucial to topological insulator as it associates the boundary states (with zero energy, chiral or helical) to topological numbers defined in bulk The application of this correspondence needs a prerequisite condition which is usually not mentioned explicitly: the boundaries themselves cannot alter the bulk states, so as to the topological numbers defined on them In non-hermitian models with fractional winding number, we prove that such precondition fails and the bulk boundary correspondence is cut out We show that, as eliminating the hopping between the boundaries to simulate the evolution of a system from the periodic boundary condition to the open boundary condition, exceptional points must be passed through and the topological structure of the spectrum has been changed This makes the topological structures of a chain with open boundary totally different from that without the boundary We also argue that such exotic behavior does not emerge when the open boundary is replaced by a domain-wall So the index theorem can be applied to the systems with domain-walls but cannot be further used to those with open boundary

Finite element quasi-interpolation and best approximation

Abstract: This paper introduces a quasi-interpolation operator for scalar-and vector-valued finite element spaces constructed on affine, shape-regular meshes with some continuity across mesh interfaces. This operator is stable in L^1 , is a projection, whether homogeneous boundary conditions are imposed or not, and, assuming regularity in the fractional Sobolev spaces W^{s,p} where p ∈ [1, ∞] and s can be arbitrarily close to zero, gives optimal local approximation estimates in any L^p-norm. The theory is illustrated on H^1-, H(curl)-and H(div)-conforming spaces.

Effects of thermal and shear deformation on vibration response of functionally graded thick composite microbeams

Abstract: In this paper, thermal and shear deformation effects on the vibrational response of non-homogeneous microbeams made of functionally graded (FG) materials are carried out. It is assumed that the temperature-dependent material properties of FG microbeams change smoothly and gradually throughout the height according to the classical rule of mixture. The governing differential equations and related boundary conditions are derived by implementing Hamilton's principle on the basis of hyperbolic shear deformation beam and modified couple stress theories and they are analytically solved. The results are given together with other beam theories. A detailed parametric study is performed to indicate the influences of slenderness ratio, material length scale parameter, gradient index, shear correction factors and temperature rise on natural frequencies of FG microbeams. It is revealed that the use of modified shear correction factor can provide more accurate and valid results for first-order shear deformable microbeam model.

Tunable Acoustic Valley-Hall Edge States in Reconfigurable Phononic Elastic Waveguides

Abstract: This study investigates the occurrence of acoustic topological edge states in a 2D phononic elastic waveguide due to a phenomenon that is the acoustic analogue of the quantum valley Hall effect. We show that a topological transition takes place between two lattices having broken space inversion symmetry due to the application of a tunable strain field. This condition leads to the formation of gapless edge states at the domain walls, as further illustrated by the analysis of the bulk-edge correspondence and of the associated topological invariants. Although time reversal symmetry is still intact in these systems, the edge states are topologically protected when inter-valley mixing is either weak or negligible. Interestingly, topological edge states can also be triggered at the boundary of a single domain if boundary conditions are properly selected. We also show that the static modulation of the strain field allows tuning the response of the material between the different supported edge states.

The Role of Fractional Time-Derivative Operators on Anomalous Diffusion

Abstract: The generalized diffusion equations with fractional order derivatives have shown be quite efficient to describe the diffusion in complex systems, with the advantage of producing exact expressions for the underlying diffusive properties. Recently, researchers have proposed different fractional-time operators (namely: the Caputo-Fabrizio and Atangana-Baleanu) which, differently from the well-known Riemann-Liouville operator, are defined by non-singular memory kernels. Here we proposed to use these new operators to generalize the usual diffusion equation. By analyzing the corresponding fractional diffusion equations within the continuous time random walk framework, we obtained waiting time distributions characterized by exponential, stretched exponential, and power-law functions, as well as a crossover between two behaviors. For the mean square displacement, we found crossovers between usual and confined diffusion, and between usual and sub-diffusion. We obtained the exact expressions for the probability distributions, where non-Gaussian and stationary distributions emerged. This former feature is remarkable because the fractional diffusion equation is solved without external forces and subjected to the free diffusion boundary conditions. We have further shown that these new fractional diffusion equations are related to diffusive processes with stochastic resetting, and to fractional diffusion equations with derivatives of distributed order. Thus, our results suggest that these new operators may be a simple and efficient way for incorporating different structural aspects into the system, opening new possibilities for modeling and investigating anomalous diffusive processes.

Natural frequencies of functionally graded plates with porosities via a simple four variable plate theory: An analytical approach

Abstract: In this paper, the free vibration analysis of rectangular plates composed of functionally graded materials with porosities is investigated based on a simple first-order shear deformation plate theory. The network of pores in assumed to be empty or filled by low pressure air and the material properties of the plate varies through the thickness. Using Hamilton's principle and utilizing the variational method, the governing equations of motion of FG plates with porosities are derived. Considering two boundary layer functions, the governing equations of the system are rewritten and decoupled. Finally, two decoupled equations are solved analytically for Levy-type boundary conditions so as to obtain the eigenfrequencies of the plate. The effects of porosity parameter, power law index, thickness-side ratio, aspect ratio, porosity distribution and boundary conditions on natural frequencies of the plate are investigated in detail.

State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter

Abstract: It is possible to describe fermionic phases of matter and spin-topological field theories in 2+1d in terms of bosonic “shadow” theories, which are obtained from the original theory by “gauging fermionic parity”. The fermionic/spin theories are recovered from their shadow by a process of fermionic anyon condensation: gauging a one-form symmetry generated by quasi-particles with fermionic statistics. We apply the formalism to theories which admit gapped boundary conditions. We obtain Turaev-Viro-like and Levin-Wen-like constructions of fermionic phases of matter. We describe the group structure of fermionic SPT phases protected by ℤ_2^f × G. The quaternion group makes a surprise appearance.

Circle compactification and ’t Hooft anomaly

Abstract: Anomaly matching constrains low-energy physics of strongly-coupled field theories, but it is not useful at finite temperature due to contamination from high-energy states. The known exception is an ’t Hooft anomaly involving one-form symmetries as in pure SU(N ) Yang-Mills theory at θ = π. Recent development about large-N volume independence, however, gives us a circumstantial evidence that ’t Hooft anomalies can also remain under circle compactifications in some theories without one-form symmetries. We develop a systematic procedure for deriving an ’t Hooft anomaly of the circle-compactified theory starting from the anomaly of the original uncompactified theory without one-form symmetries, where the twisted boundary condition for the compactified direction plays a pivotal role. As an application, we consider $$ {\mathbb{Z}}_N $$ -twisted $$ \mathbb{C}{P}^{N-1} $$ sigma model and massless $$ {\mathbb{Z}}_N $$ -QCD, and compute their anomalies explicitly.

Coronal Magnetic Field Models

Abstract: Coronal magnetic field models use photospheric field measurements as boundary condition to model the solar corona. We review in this paper the most common model assumptions, starting from MHD-models, magnetohydrostatics, force-free and finally potential field models. Each model in this list is somewhat less complex than the previous one and makes more restrictive assumptions by neglecting physical effects. The magnetohydrostatic approach neglects time-dependent phenomena and plasma flows, the force-free approach neglects additionally the gradient of the plasma pressure and the gravity force. This leads to the assumption of a vanishing Lorentz force and electric currents are parallel (or anti-parallel) to the magnetic field lines. Finally, the potential field approach neglects also these currents. We outline the main assumptions, benefits and limitations of these models both from a theoretical (how realistic are the models?) and a practical viewpoint (which computer resources to we need?). Finally we address the important problem of noisy and inconsistent photospheric boundary conditions and the possibility of using chromospheric and coronal observations to improve the models.