Merging Discrete Morse Vector Fields: A Case of Stubborn Geometric Parallelization

We address the basic question in discrete Morse theory of combining discrete gradient fields that are partially defined on subsets of the given complex. This is a well-posed question when the discrete gradient field V is generated using a fixed algorithm which has a local nature. One example is ProcessLowerStars, a widely used algorithm for computing persistent homology associated to a grey-scale image in 2D or 3D. While the algorithm for V may be inherently local, being computed within stars of vertices and so embarrassingly parallelizable, in practical use, it is natural to want to distribute the computation over patches Pi, apply the chosen algorithm to compute the fields Vi associated to each patch, and then assemble the ambient field V from these. Simply merging the fields from the patches, even when that makes sense, gives a wrong answer. We develop both very general merging procedures and leaner versions designed for specific, easy-to-arrange covering patterns.

A FORTRAN Program to Model Magnetic Gradient Tensor at High Susceptibility Using Contraction Integral Equation Method

The magnetic gradient tensor provides a powerful tool for detecting magnetic bodies because of its ability to emphasize detailed features of the magnetic anomalies. To interpret field measurements obtained by magnetic gradiometry, the forward calculation of magnetic gradient fields is always necessary. In this paper, we present a contraction integral equation method to simulate the gradient fields produced by 3-D magnetic bodies of arbitrary shapes and high susceptibilities. The method employs rectangular prisms to approximate the source region with the assumption that the magnetization in each element is homogeneous. The gradient fields are first solved in the Fourier domain and then transformed into the spatial domain by 2-D Gauss-FFT. This calculation is performed iteratively until the required accuracy is reached. The convergence of the iterative procedure is ensured by a contraction operator. To facilitate application, we introduce a FORTRAN program to implement the algorithm. This program is intended for users who show interests in 3D magnetic modeling at high susceptibility. The performance of the program, including its computational accuracy, efficiency and convergence behavior, is tested by several models. Numerical results show that the code is computationally accurate and efficient, and performs well at a wide range of magnetic susceptibilities from 0 SI to 1000 SI. This work, therefore, provides a significant tool for 3D forward modeling of magnetic gradient fields at high susceptibility.

Study of inhomogeneous fields of residual stresses using step-by-step enlarged crack method in combination with electronic speckle pattern interferometry (ESPI)

This article covers the development of methodological issues, software, experimental equipment, and practical application of the method of sequentially increasing cracks for studying inhomogeneous high-gradient fields of residual stresses (RS) that occur in areas of structural heterogeneity in flat construction parts (e.g., welded joints). Method of electronic speckle pattern interferometry (ESPI) is used to detect the deformation response in the form of the fields of displacement of the surface of the object under study arising from formation of successively increasing crack. ESPI provide contactless registration of movements directly in a digital form with high accuracy. The scheme of a specialized interferometer is described along with the features of the procedure for registering the displacement fields arising from a stepwise increase of the crack length. A return device provided removing of the object under study out of the optical zone and then return it to the initial position after performing the necessary mechanical operations. The accuracy of the procedure for calculating RS is estimated on the basis of mathematical processing of the experimentally obtained dependences of SIF on the crack length. An example of using the developed methods, equipment and programs for studying the RS distribution in stir welding joint of the sheets of aircraft alloy 1163T with a high level of crack resistance is given.

Locomotion and disaggregation control of paramagnetic nanoclusters using wireless electromagnetic fields for enhanced targeted drug delivery

Magnetic nanorobots (MNRs) based on paramagnetic nanoparticles/nanoclusters for the targeted therapeutics of anticancer drugs have been highlighted for their efficiency potential. Controlling the locomotion of the MNRs is a key challenge for effective delivery to the target legions. Here, we present a method for controlling paramagnetic nanoclusters through enhanced tumbling and disaggregation motions with a combination of rotating field and gradient field generated by external electromagnets. The mechanism is carried out via an electromagnetic actuation system capable of generating MNR motions with five degrees of freedom in a spherical workspace without singularity. The nanocluster swarm structures can successfully pass through channels to the target region where they can disaggregate. The results show significantly faster response and higher targeting rate by using rotating magnetic and gradient fields. The mean velocities of the enhanced tumbling motion are twice those of the conventional tumbling motion and approximately 130% higher than the gradient pulling motion. The effects of each fundamental factor on the locomotion are investigated for further MNR applications. The locomotion speed of the MNR could be predicted by the proposed mathematical model and agrees well with experimental results. The high access rate and disaggregation performance insights the potentials for targeted drug delivery application.

Convolutional Neural Networks for the Localization of Plastic Velocity Gradient Tensor in Polycrystalline Microstructures

Recent work has demonstrated the potential of convolutional neural networks (CNNs) in producing low-computational cost surrogate models for the localization of mechanical fields in two-phase microstructures. The extension of the same CNNs to polycrystalline microstructures is hindered by the lack of an efficient formalism for the representation of the crystal lattice orientation in the input channels of the CNNs. In this paper, we demonstrate the benefits of using generalized spherical harmonics (GSH) for addressing this challenge. A CNN model was successfully trained to predict the local plastic velocity gradient fields in polycrystalline microstructures subjected to a macroscopically imposed loading condition. Specifically, it is demonstrated that the proposed approach improves significantly the accuracy of the CNN models, when compared with the direct use of Bunge-Euler angles to represent the crystal orientations in the input channels. Since the proposed approach implicitly satisfies the expected crystal symmetries in the specification of the input microstructure to the CNN, it opens new research directions for the adoption of CNNs in addressing a broad range of polycrystalline microstructure design and optimization problems.

Pressure-robust error estimate of optimal order for the Stokes equations: domains with re-entrant edges and anisotropic mesh grading

The velocity solution of the incompressible Stokes equations is not affected by changes of the right hand side data in form of gradient fields. Most mixed methods do not replicate this property in the discrete formulation due to a relaxation of the divergence constraint which means that they are not pressure-robust. A recent reconstruction approach for classical methods recovers this invariance property for the discrete solution, by mapping discretely divergence-free test functions to exactly divergence-free functions in the sense of $${\varvec{H}}({\text {div}})$$ H ( div ) . Moreover, the Stokes solution has locally singular behavior in three-dimensional domains near concave edges, which degrades the convergence rates on quasi-uniform meshes and makes anisotropic mesh grading reasonable in order to regain optimal convergence characteristics. Finite element error estimates of optimal order on meshes of tensor-product type with appropriate anisotropic grading are shown for the pressure-robust modified Crouzeix–Raviart method using the reconstruction approach. Numerical examples support the theoretical results.