Conforming boundary element methods in acoustics

In developing accurate and stable discretization schemes for integral equations, important steps are the choices of the basis and testing functions. In the electromagnetic surface integral equations, the unknown quantities, the equivalent electric and magnetic surface current densities, are by default divergence conforming and therefore should be expanded with the divergence‐conforming basis functions. The choice of the testing functions however is less obvious and depends on the choice of the basis functions and the properties of the integral operators. We develop stable and well‐defined discretization procedures where the operators are tested either their range or on the dual of their range. This requires the use of non‐conventional finite‐elements spaces, called ‘dual spaces’, and non‐traditional testing procedures. We show with numerical examples that in most cases, testing the operators on the dual of their range leads to improved solution accuracy compared with testing the operators on their range. For the integral equations of the first kind, this agrees with the conventional Galerkin's scheme, but for the integral equations of the second kind, this requires the use of Petrov–Galerkin type discretization schemes with appropriate curl‐conforming (dual) testing functions. Copyright © 2012 John Wiley & Sons, Ltd.

The rapid advancements in robotics and automation have highlighted the need for robotic systems capable of adapting to the diverse physical properties of objects. Traditional grippers often lack the versatility to handle both hard and soft objects without extensive reprogramming or hardware adjustments. Previous studies have explored various approaches to this challenge, including tactile sensors and force feedback mechanisms to distinguish object properties. For instance, research by Calandra et al. (2018) utilized deep learning with tactile data to enable robotic hands to identify objects and adjust grip accordingly. Similarly, the paper by Li et al. (2020) “Design and performance characterization of a soft robot hand with fingertip haptic feedback for teleoperation”, focuses on designing and characterizing a soft robotic hand with fingertip haptic feedback for teleoperation emphasizing real-time tactile sensing and feedback mechanisms. However, these studies primarily focus on tactile feedback or specialized hardware, limiting their applicability in scenarios where such systems are not available or practical.This study introduces a novel machine learning-based approach, focusing on the use of visual data alone to classify and adapt to the hardness or softness of objects. By leveraging the CIFAR-100 dataset, we trained a deep learning model based on the ResNet50 architecture, achieving significant results in binary classification of hard and soft objects. The CIFAR-100 dataset, consisting of 100 diverse object categories/classes, was reorganized into two classes: hard (39 categories) and soft (61 categories). The ResNet50 model, pre-trained on ImageNet, was fine-tuned specifically for this task, with modifications to the last 210 layers to enhance its adaptability.Data augmentation techniques, including rotations, translations, shearing, zooming, and horizontal flipping, were applied to simulate real-world variations, ensuring robust learning. The model was further refined with additional fully connected layers, dropout, and batch normalization to prevent overfitting. Optimized using the AdamW optimizer, the model achieved a training accuracy of 83.31% and a validation accuracy of 80.25%, with a test accuracy of 80%. The precision, recall, and F1-scores were 0.82, 0.86, and 0.84 for the soft object class, and 0.76, 0.71, and 0.73 for the hard object class, demonstrating the model’s effectiveness in distinguishing between hard and soft objects without the need for specialized sensors. It is also observed that the accuracy of the model in relation to hard objects is significantly lesser as compared to soft objects. It is understood that the CIFAR-100 dataset (comprising of 100 classes) is inadequate for model training, so we are exploring the ILSVRC (ImageNet subset) dataset for model training in future. The research is useful in a variety of fields where the focus lies on object handling. In everyday life, we encounter a wide array of objects with varying degrees of hardness, requiring different levels of care and precision during handling. By relying on visual data and machine learning algorithms, as demonstrated in this research, robotic systems can become more autonomous and versatile, reducing the burden on human operators and improving overall efficiency. This approach can lead to cost savings by reducing the need for specialized hardware, such as tactile sensors, which are often expensive and difficult to integrate.

Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems

This chapter presents a treatment of the mathematical aspects of continuum elastoplasticity. The focus is on rate‐independent material behavior, with the flow relations constructed within the framework of convex analysis. The basis for variational treatments of the problem is the versions of the flow relation that use the dissipation function or, alternatively, the yield surface with a normality law: these are referred to, respectively, as the primal and dual formulations. Both formulations are discussed, with emphasis on the former. Some results on the convergence of fully discrete approximations, using finite elements in space, are presented. The associated predictor–corrector algorithms are described, and their convergence behavior is discussed. A discussion of the large‐deformation theory includes an overview of the derivative‐free energetic formulation, due to Mielke and coworkers, and of which the primal formulation discussed in the small‐strain theory is a special case. Relevant results from functional analysis and function spaces are summarized at appropriate places in the presentation.

Exponentially accurate spectral and spectral element methods for fractional ODEs

By means of the symbolic manipulation language REDUCE, a complete set of harmonic polynomials in Euclidean space is constructed. The method relies on the theory of monogenic functions defined in ℝn+1 and taking values in a Clifford algebra An. An approximate solution for the Dirichlet and Neumann problems in the form of a harmonic polynomial may be obtained. The method is applied to solying a Dirichlet problem in a cube.

We propose a reconstruction-based a posteriori error estimate for linear advection problems in one space dimension. In our framework, a stable variational ultra-weak formulation is adopted, and the equivalence of the L2-norm of the error with the dual graph norm of the residual is established. This dual norm is showed to be localizable over vertex-based patch subdomains of the computational domain under the condition of the orthogonality of the residual to the piecewise affine hat functions. We show that this condition is valid for some well-known numerical methods including continuous/discontinuous Petrov–Galerkin and discontinuous Galerkin methods. Consequently, a well-posed local problem on each patch is identified, which leads to a global conforming reconstruction of the discrete solution. We prove that this reconstruction provides a guaranteed upper bound on the L2 error. Moreover, up to a generic constant, it also gives local lower bounds on the L2 error, where the constant only depends on the mesh shape-regularity. This, in particular, leads to robustness of our estimates with respect to the polynomial degree. All the above properties are verified in a series of numerical experiments, additionally leading to asymptotic exactness. Motivated by these results, we finally propose a heuristic extension of our methodology to any space dimension, achieved by solving local least-squares problems on vertex-based patches. Though not anymore guaranteed, the resulting error indicator is still numerically robust with respect to both advection velocity and polynomial degree in our collection of two-dimensional test cases including discontinuous solutions aligned and not aligned with the computational mesh.

We study the Neumann and Dirichlet problems for the total variation flow in doubling metric measure spaces supporting a weak Poincaré inequality. We prove existence and uniqueness of weak solutions and study their asymptotic behavior. Furthermore, in the Neumann problem we provide a notion of solutions which is valid for L 1 {L^{1}} initial data, as well as prove their existence and uniqueness. Our main tools are the first-order linear differential structure due to Gigli and a version of the Gauss–Green formula.

The authors introduce a new method for the Dirichlet and Neumann problems (in \mathbb R^2 and \mathbb R^3 ) based on certain theorems of Zaremba and Bergmann. With the help of a symbolic computing language (FORMAC with PL/1), the authors show how it is possible to orthonormalize a complete sequence of harmonic polynomials on the domain G of the solution. The solution is then given by an expansion in these orthonormalized polynomials. Under certain conditions on the boundary values, the expansion coefficients themselves may also be automatically calculated.

This article is devoted to the study of a fully discrete A‐ finite element method to solve nonlinear Maxwell's equations based on backward Euler discretization in time and nodal finite elements in space. The nonlinearity is owing to a field‐dependent conductivity with the power‐law form . We design a nonlinear time‐discrete scheme for approximation in suitable function spaces. We show the well‐posedness of the problem, prove the convergence of the semidiscrete scheme based on the boundedness of the second derivative in the dual space and derive its error estimate. The Minty–Browder technique is introduced to obtain the convergence of the nonlinear term. Finally, we discuss the error estimate for the fully discretized problem and support the theoretical result by two numerical experiments. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 2083–2108, 2014

A Petrov–Galerkin spectral element method for fractional elliptic problems

By analogy with the Green function for the Dirichlet problem for a domain , one considers the Green type function for the Neumann problem for . We obtain the Green type function for the positive half-space of R n and use it to solve the Neumann problem. The cases where is the positive half space of R 2 and of R 3 are analyzed in details under new hypotheses.

A discontinuous Petrov–Galerkin methodology for adaptive solutions to the incompressible Navier–Stokes equations

. In this article, the interval expansion of the structure of solving basic types of boundary value problems for partial differential equations of the second order of making the basic operations that compose interval arithmetic is developed. For the differential equation (1) of the type, when constructing the interval expansion of the structure of the formula, structural formulas were used to construct with the Rfunction method and 4 problems were studied — the Dirichlet problem, the Neumann problem, the third type problem, the mixed boundary conditions problem. For the Dirichlet problem, the solution is an interval expansion of the structure in the form (5), where 𝑃 = {𝜔𝛷 , 𝜔𝛷̅, 𝜔̅𝛷, 𝜔̅𝛷̅} и [ 𝛷, 𝛷̅]is an indefinite interval function. For the Neumann problem, a solution is solved in the interval extension of the structure, [ 𝛷1, 𝛷1̅̅̅̅], [ 𝛷2, 𝛷2̅̅̅̅] is an indefinite interval function and 𝐷1 is a differential operator of the form. For the problem of the third type, the solution is solved in the interval extension of the structure, [ 𝛷1, 𝛷1̅̅̅̅], [𝛷2, 𝛷2̅̅̅̅] -indefinite, interval function, 𝐷1 - differential operator of the form (9). For the problem, mixed boundary conditions are treated. The solution In the interval extension of the structure,[ 𝛷1, 𝛷1], [ 𝛷2, 𝛷2̅̅̅̅] is an indefinite interval function and 𝐷1 is a differential operator of the form.

This is the first part of a series of two papers where we study perturbations of divergence form second order elliptic operators $-\mathop{\operatorname{div}} A \nabla$ by first and zero order terms, whose coefficients lie in critical spaces, via the method of layer potentials. In particular, we show that the $L^2$ well-posedness of the Dirichlet, Neumann and Regularity problems for complex Hermitian, block form, or constant-coefficient divergence form elliptic operators in the upper half-space are all stable under such perturbations. For instance, this allows us to claim the first results in the setting of an unbounded domain concerning the solvability of boundary value problems for the magnetic Schr\"odinger operator $-(\nabla-i{\bf a})^2+V$ when the magnetic potential ${\bf a}$ and the electric potential $V$ are accordingly small in the norm of a scale-invariant Lebesgue space. In the present paper, we establish $L^2$ control of the square function via a vector-valued $Tb$ theorem and abstract layer potentials, and use these square function bounds to obtain uniform slice bounds for solutions. The existence and uniqueness of solutions, as well as bounds for the non-tangential maximal operator, are considered in the upcoming paper.