Pointwise Sense Research Articles

Second-Order Two-Scale Computational Method for Nonlinear Dynamic Thermo-Mechanical Problems of Composites with Cylindrical Periodicity

AbstractIn this paper, a novel second-order two-scale (SOTS) computational method is developed for nonlinear dynamic thermo-mechanical problems of composites with cylindrical periodicity. The non-linearities of these multi-scale problems were caused by the temperature-dependent properties of the composites. Firstly, the formal SOTS solutions for these problems are constructed by the multiscale asymptotic analysis. Then we theoretically explain the importance of the SOTS solutions by the error analysis in the pointwise sense. In addition, a SOTS numerical algorithm is proposed in detail to effectively solve these problems. Finally, some numerical examples verify the feasibility and effectiveness of the SOTS numerical algorithm we proposed.

Comparison of Two Galerkin Quadrature Methods

We compare two methods for generating Galerkin quadratures. In method 1, the standard SN method is used to generate the moment-to-discrete matrix and the discrete-to-moment matrix is generated by inverting the moment-to-discrete matrix. This is a particular form of the original Galerkin quadrature method. In method 2, which we introduce here, the standard SN method is used to generate the discrete-to-moment matrix and the moment-to-discrete matrix is generated by inverting the discrete-to-moment matrix. With an N-point quadrature, method 1 has the advantage that it preserves N eigenvalues and N eigenvectors of the scattering operator in a pointwise sense. With an N-point quadrature, method 2 has the advantage that it generates consistent angular moment equations from the corresponding SN equations while preserving N eigenvalues of the scattering operator. Our computational results indicate that these two methods are quite comparable for the test problem considered.

Second-order two-scale analysis and numerical algorithm for the damped wave equations of composite materials with quasi-periodic structures

In this paper, we perform a second-order two-scale analysis and introduce a numerical algorithm for the damped wave equations of composite materials with a quasi-periodic structure. Firstly, second-order two-scale asymptotic expansion solutions for these problems are constructed by a multiscale asymptotic analysis. In addition, we explain the importance of the second-order two-scale solutions by the error analysis in the pointwise sense. Moreover, explicit convergence rates of these second-order two-scale solutions are obtained in the integral sense. Then a second-order two-scale numerical method based on a Newmark scheme is presented to solve these multiscale problems. Finally, some numerical examples show the effectiveness and efficiency of the multiscale numerical method we proposed.

Iterative Learning Control for discrete nonlinear systems with randomly iteration varying lengths

This note proposes ILC for discrete-time affine nonlinear systems with randomly iteration varying lengths. No prior information on the probability distribution of random iteration length is required prior for controller design. The conventional P-type update law is used with a modified tracking error because of randomly iteration varying lengths. A novel technical lemma is proposed for the strict convergence analysis in pointwise sense. An illustrative example verifies the theoretical results.

Generalized Poisson Summation Formulas for Continuous Functions of Polynomial Growth

The Poisson summation formula (PSF) describes the equivalence between the sampling of an analog signal and the periodization of its frequency spectrum. In engineering textbooks, the PSF is usually stated formally without explicit conditions on the signal for the formula to hold. By contrast, in the mathematics literature, the PSF is commonly stated and proven in the pointwise sense for various types of \(L_1\) signals. This \(L_1\) assumption is, however, too restrictive for many signal-processing tasks that demand the sampling of possibly growing signals. In this paper, we present two generalized versions of the PSF for d-dimensional signals of polynomial growth. In the first generalization, we show that the PSF holds in the space of tempered distributions for every continuous and polynomially growing signal. In the second generalization, the PSF holds in a particular negative-order Sobolev space if we further require that \(d/2+\varepsilon \) derivatives of the signal are bounded by some polynomial in the \(L_2\) sense.

Categorical representation of North American precipitation projections.

We explore use of the familiar tercile framework of seasonal forecasting for the characterization of 21st-century precipitation projections over North America. Consistent with direct analyses of modeled precipitation change, in a superensemble of CMIP5 simulations an unambiguous pattern of shifted tercile population statistics develops as the globe warms. Expressed categorically, frequencies for the low (i.e., dry) tercile increase in the southwestern United States and southward into Mexico and decrease across the northern tier of North America, while counts for the high tercile shift in the opposite sense. We show that as the 21st-century proceeds, changes become statistically significant over wide regions in the pointwise sense, and also when considered as projections on model-specific climate change “fingerprints”. Background noise in the superensemble, against which significance is established, comprises both structural model uncertainty and natural climate variability. The robustness of these findings makes a compelling case for long-range planning for a dryer future in the American Southwest and southward, and wetter one to the north and especially northeast, while communication is facilitated by widespread user familiarity with the tercile format.

Pointwise time asymptotic behavior of solutions to a general class of hyperbolic balance laws

In this paper we are interested in a general class of hyperbolic balance laws. Within the frame work of existence of a convex entropy function that symmetrizes the system in certain sense and the Kawashima–Shizuta condition, we study the large time behavior of the solution to the Cauchy problem in one space dimension. For a solution around a constant equilibrium state, we predetermine the asymptotic solution as a superposition of the constant state and diffusion waves along the equilibrium characteristic directions. We estimate the remainder in the pointwise sense both in space and in time. The decay rates in various directions are optimal, which further give the optimal Lp rates with 1≤p≤∞. Applications to several examples including the Kerr–Debye model are given.

Some characterizations on weighted $$\beta \gamma $$ β γ -statistical convergence of fuzzy functions of order $$\theta $$ θ

Based on the concept of new type of statistical convergence defined by Aktuglu, we have introduced the weighted $\alpha\beta$ - statistical convergence of order $\theta$ in case of fuzzy functions and classified it into pointwise, uniform and equi-statistical convergence. We have checked some basic properties and then the convergence are investigated in terms of their $\alpha$-cuts. The interrelation among them are also established. We have also proved that continuity, boundedness etc are preserved in the equi-statistical sense under some suitable conditions, but not in pointwise sense.

Improving the Accuracy of Registration-Based Biomechanical Analysis: A Finite Element Approach to Lung Regional Strain Quantification.

Tissue deformation plays an important role in lung physiology, as lung parenchyma largely deforms during spontaneous ventilation. However, excessive regional deformation may lead to lung injury, as observed in patients undergoing mechanical ventilation. Thus, the accurate estimation of regional strain has recently received great attention in the intensive care community. In this work, we assess the accuracy of regional strain maps computed from direct differentiation of B-Spline (BS) interpolations, a popular technique employed in non-rigid registration of lung computed tomography (CT) images. We show that, while BS-based registration methods give excellent results for the deformation transformation, the strain field directly computed from BS derivatives results in predictions that largely oscillate, thus introducing important errors that can even revert the sign of strain. To alleviate such spurious behavior, we present a novel finite-element (FE) method for the regional strain analysis of lung CT images. The method follows from a variational strain recovery formulation, and delivers a continuous approximation to the strain field in arbitrary domains. From analytical benchmarks, we show that the FE method results in errors that are a fraction of those found for the BS method, both in an average and pointwise sense. The application of the proposed FE method to human lung CT images results in 3D strain maps are heterogeneous and smooth, showing high consistency with specific ventilation maps reported in the literature. We envision that the proposed FE method will considerably improve the accuracy of image-based biomechanical analysis, making it reliable enough for routine medical applications.

Second-order two-scale analysis and numerical algorithms for the hyperbolic–parabolic equations with rapidly oscillating coefficients**Project supported by the National Natural Science Foundation of China (Grant No. 11471262), the National Basic Research Program of China (Grant No. 2012CB025904), and the State Key Laboratory of Science and Engineering Computing and the Center for High Performance Computing of Northwestern Polytechnical University, China.

We study the hyperbolic–parabolic equations with rapidly oscillating coefficients. The formal second-order two-scale asymptotic expansion solutions are constructed by the multiscale asymptotic analysis. In addition, we theoretically explain the importance of the second-order two-scale solution by the error analysis in the pointwise sense. The associated explicit convergence rates are also obtained. Then a second-order two-scale numerical method based on the Newmark scheme is presented to solve the equations. Finally, some numerical examples are used to verify the effectiveness and efficiency of the multiscale numerical algorithm we proposed.

The continuous wavelet transform and window functions

We define a window function ψ \psi as an element of L 2 ( R n ) L^2(\mathbb R^n) satisfying certain boundedness properties with respect to the L 2 ( R n ) L^2(\mathbb R^n) norm and prove that it satisfies the admissibility condition if and only if the integral of ψ ( x 1 , x 2 , ⋯ , x n ) \psi (x_1,x_2,\cdots ,x_n) with respect to each of the variables x 1 , x 2 , ⋯ , x n x_1,x_2,\cdots ,x_n along the real line is zero. We also prove that each of the window functions is an element of L 1 ( R n ) L^1(\mathbb R^n) . A function ψ ∈ L 2 ( R n ) \psi \in L^2(\mathbb R^n) satisfying the admissibility condition is a wavelet. We define the wavelet transform of f ∈ L 2 ( R n ) f\in L^2(\mathbb R^n) (which is a window function) with respect to the wavelet ψ ∈ L 2 ( R n ) \psi \in L^2(\mathbb R^n) and prove an inversion formula interpreting convergence in L 2 ( R n ) L^2(\mathbb R^n) . It is also proved that at a point of continuity of f f the convergence of our wavelet inversion formula is in a pointwise sense.

Indentation of sandwiches using a plate model with variable kinematics and fixed degrees of freedom

A recent 3-D, 5 degrees of freedom, variable kinematics zig-zag homogenised model is used for studying sandwich indentation. This model cannot describe in a point-wise sense cells collapse, buckling, debonding failure, plastic yielding, or brittle fracture. Therefore, the progressive folding followed by densification of the core is described through mechanical properties variable with loading, which are computed apart once at a time through a 3D FEA discretisation of the core structure. Onset of damage is determined using stress-based criteria; degraded properties of failed regions are obtained from a mesoscale damage model. The results show efficiency and accuracy of this approach.

Second-order two-scale method for bending behavior analysis of composite plate with 3-D periodic configuration and its approximation

This paper considers the bending behaviors of composite plate with 3-D periodic configuration. A second-order two-scale (SOTS) computational method is designed by means of construction way. First, by 3-D elastic composite plate model, the cell functions which are defined on the reference cell are constructed. Then the effective homogenization parameters of composites are calculated, and the homogenized plate problem on original domain is defined. Based on the Reissner-Mindlin deformation pattern, the homogenization solution is obtained. And then the SOTS’s approximate solution is obtained by the cell functions and the homogenization solution. Second, the approximation of the SOTS’s solution in energy norm is analyzed and the residual of SOTS’s solution for 3-D original in the pointwise sense is investigated. Finally, the procedure of SOTS’s method is given. A set of numerical results are demonstrated for predicting the effective parameters and the displacement and strains of composite plate. It shows that SOTS’s method can capture the 3-D local behaviors caused by 3-D micro-structures well.

On Second-Order Differentiability with Respect to Parameters for Differential Equations with State-Dependent Delays

In this paper we consider a class of differential equations with state-dependent delays. We show first and second-order differentiability of the solution with respect to parameters in a pointwise sense and also using the \(C\)-norm on the state-space, assuming that the state-dependent time lag function is piecewise strictly monotone.

Asymptotics of the Solutions of the Stochastic Lattice Wave Equation

We consider the long time limit theorems for the solutions of a discrete wave equation with a weak stochastic forcing. The multiplicative noise conserves the energy and the momentum. We obtain a time-inhomogeneous Ornstein-Uhlenbeck equation for the limit wave function that holds both for square integrable and statistically homogeneous initial data. The limit is understood in the point-wise sense in the former case, and in the weak sense in the latter. On the other hand, the weak limit for square integrable initial data is deterministic.

Relations Between 𝓛<sup><em>p</em></sup>- and Pointwise Convergence of Families of Functions Indexed by the Unit Interval

We construct a variety of mappings from the unit interval \(\mathcal{I}\) into \(\mathcal{L}^p([0,1]),1\leq p<\infty,\) to generalize classical examples of \(\mathcal{L}^p\)-converging sequences of functions with simultaneous pointwise divergence. By establishing relations between the regularity of the functions in the image of the mappings and the topology of \(\mathcal{I}\), we obtain examples which are \(\mathcal{L}^p\)-continuous but exhibit discontinuity in a pointwise sense to different degrees. We conclude by proving a Lusin-type theorem, namely that if almost every function in the image is continuous, then we can remove a set of arbitrarily small measure from the index set \(\mathcal{I}\) and establish pointwise continuity in the remainder.

Superconvergence of tetrahedral quadratic finite elements for a variable coefficient elliptic equation

AbstractFor a variable coefficient elliptic boundary value problem in three dimensions, using the properties of the bubble function and the element cancelation technique, we derive the weak estimate of the first type for tetrahedral quadratic elements. In addition, the estimate for the W1,1‐seminorm of the discrete derivative Green's function is also given. Finally, we show that the derivatives of the finite element solution uh and the corresponding interpolant Π2u are superclose in the pointwise sense of the L∞‐norm. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

Eccentricity effects in the finite element modelling of composite beams

When modelling composite or built up beams using finite element software, analysts find it often convenient to connect two standard Euler-Bernoulli beam elements at the nodes by using a rigid bar or use master-slave type kinematic constraints to express the degrees-of-freedoms of one of the members in terms of the other. However, this type of modelling leads to eccentricity related numerical errors and special solutions that avoid eccentricity related issues may not be available for a design engineer due to the limitations of the software. In this study, a simple correction technique is introduced in the application of master-slave type constraints. It is shown that the eccentricity related numerical errors in the stiffness matrix can be completely corrected by using extra fictitious elements and springs. The correction terms are obtained by using the exact homogenous solution of the composite beam problem as the interpolation functions which impose the zero-slip constraint between the two components in the point-wise sense. The effects of the eccentricity related errors are demonstrated in numerical examples.

Variational multiscale approach to enforce perfect bond in multiple-point constraint applications when forming composite beams

Composite laminates that consist of two or more layers find widespread applications in a variety of engineering structures. In the computational modelling of composite laminates, the layers can be stacked together and connected conveniently at the nodes by using multiple-point constraints (MPCs). However, this type of modelling leads to weakening of the kinematic constraint conditions imposed by the bond between the juxtaposed layers and as a consequence, MPCs application at the nodes produces behaviour that is softer than the perfectly bonded composite beam behaviour. The work herein shows that when kinematic conditions for composite action are weakly imposed in the variational form, they can be enforced in the point-wise sense by proper selection of the interpolation field or otherwise reinforced by using variational multiscale approach without modifying the kinematic model. The originality of the approach presented herein is in the interpretation of the MPCs application as the solution in a superfluously extended space because of the weakening in the kinematic constraints. It is shown that the perfect bond between the composite beam layers can be recovered by excluding the identified fine-scale effect from the solution of the multiple point constraint application. The convergence characteristic of the finite element formulation is also improved by using the variational multi-scale approach. It is also shown that the fine-scale effects can be represented by using extra fictitious elements and springs, which offers a direct correction technique in modelling of composite beams that is especially useful when access to the numerical procedure is limited.

On differentiability of solutions with respect to parameters in neutral differential equations with state-dependent delays

In this paper, we consider a class of nonlinear neutral differential equations with state-dependent delays in both the neutral and the retarded terms. We study well-posedness and continuous dependence issues and differentiability of the parameter map with respect to the initial function and other possibly infinite dimensional parameters in a pointwise sense and also in the C-norm.