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Python serves as an open-source and cost-effective alternative to the MATLAB programming language. This paper introduces a concise topology optimization Python code, named ``\texttt{PyHexTop}," primarily intended for educational purposes. Code employs hexagonal elements to parameterize design domains as such elements provide checkerboard-free optimized design naturally. \texttt{PyHexTop} is developed based on the ``\texttt{HoneyTop90}" MATLAB code~\cite{kumar2023honeytop90} and uses the \texttt{NumPy} and \texttt{SciPy} libraries. Code is straightforward and easily comprehensible, proving a helpful tool that can help people new in the topology optimization field to learn and explore. \texttt{PyHexTop} is specifically tailored to address compliance minimization with specified volume constraints. The paper provides a detailed explanation of the code for solving the Messerschmitt-Bolkow-Blohm beam and extensions to solve problems different problems. The code is publicly shared at: \url{//github.com/PrabhatIn/PyHexTop.}

We present a complete numerical analysis for a general discretization of a coupled flow-mechanics model in fractured porous media, considering single-phase flows and including frictionless contact at matrix-fracture interfaces, as well as nonlinear poromechanical coupling. Fractures are described as planar surfaces, yielding the so-called mixed- or hybrid-dimensional models. Small displacements and a linear elastic behavior are considered for the matrix. The model accounts for discontinuous fluid pressures at matrix-fracture interfaces in order to cover a wide range of normal fracture conductivities. The numerical analysis is carried out in the Gradient Discretization framework, encompassing a large family of conforming and nonconforming discretizations. The convergence result also yields, as a by-product, the existence of a weak solution to the continuous model. A numerical experiment in 2D is presented to support the obtained result, employing a Hybrid Finite Volume scheme for the flow and second-order finite elements ($\mathbb P_2$) for the mechanical displacement coupled with face-wise constant ($\mathbb P_0$) Lagrange multipliers on fractures, representing normal stresses, to discretize the contact conditions.

It is well known since 1960s that by exploring the tensor product structure of the discrete Laplacian on Cartesian meshes, one can develop a simple direct Poisson solver with an $\mathcal O(N^{\frac{d+1}d})$ complexity in d-dimension, where N is the number of the total unknowns. The GPU acceleration of numerically solving PDEs has been explored successfully around fifteen years ago and become more and more popular in the past decade, driven by significant advancement in both hardware and software technologies, especially in the recent few years. We present in this paper a simple but extremely fast MATLAB implementation on a modern GPU, which can be easily reproduced, for solving 3D Poisson type equations using a spectral-element method. In particular, it costs less than one second on a Nvidia A100 for solving a Poisson equation with one billion degree of freedoms. We also present applications of this fast solver to solve a linear (time-independent) Schr\"odinger equation and a nonlinear (time-dependent) Cahn-Hilliard equation.

In this paper, we formulate, analyse and implement the discrete formulation of the Brinkman problem with mixed boundary conditions, including slip boundary condition, using the Nitsche's technique for virtual element methods. The divergence conforming virtual element spaces for the velocity function and piecewise polynomials for pressure are approached for the discrete scheme. We derive a robust stability analysis of the Nitsche stabilized discrete scheme for this model problem. We establish an optimal and vigorous a priori error estimates of the discrete scheme with constants independent of the viscosity. Moreover, a set of numerical tests demonstrates the robustness with respect to the physical parameters and verifies the derived convergence results.

We introduce a sparse estimation in the ordinary kriging for functional data. The functional kriging predicts a feature given as a function at a location where the data are not observed by a linear combination of data observed at other locations. To estimate the weights of the linear combination, we apply the lasso-type regularization in minimizing the expected squared error. We derive an algorithm to derive the estimator using the augmented Lagrange method. Tuning parameters included in the estimation procedure are selected by cross-validation. Since the proposed method can shrink some of the weights of the linear combination toward zeros exactly, we can investigate which locations are necessary or unnecessary to predict the feature. Simulation and real data analysis show that the proposed method appropriately provides reasonable results.

Nonparametric estimators for the mean and the covariance functions of functional data are proposed. The setup covers a wide range of practical situations. The random trajectories are, not necessarily differentiable, have unknown regularity, and are measured with error at discrete design points. The measurement error could be heteroscedastic. The design points could be either randomly drawn or common for all curves. The estimators depend on the local regularity of the stochastic process generating the functional data. We consider a simple estimator of this local regularity which exploits the replication and regularization features of functional data. Next, we use the ``smoothing first, then estimate'' approach for the mean and the covariance functions. They can be applied with both sparsely or densely sampled curves, are easy to calculate and to update, and perform well in simulations. Simulations built upon an example of real data set, illustrate the effectiveness of the new approach.

This paper analyzes a full discretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. The discretization uses the Euler scheme for temporal discretization and the finite element method for spatial discretization. A key contribution of this work is the introduction of a novel stability estimate for a discrete stochastic convolution, which plays a crucial role in establishing pathwise uniform convergence estimates for fully discrete approximations of nonlinear stochastic parabolic equations. By using this stability estimate in conjunction with the discrete stochastic maximal $L^p$-regularity estimate, the study derives a pathwise uniform convergence rate that concerns the general spatial $L^q$-norms.

Lattice structures have been widely used in applications due to their superior mechanical properties. To fabricate such structures, a geometric processing step called triangulation is often employed to transform them into the STL format before sending them to 3D printers. Because lattice structures tend to have high geometric complexity, this step usually generates a large amount of triangles, a memory and compute-intensive task. This problem manifests itself clearly through large-scale lattice structures that have millions or billions of struts. To address this problem, this paper proposes to transform a lattice structure into an intermediate model called meta-mesh before undergoing real triangulation. Compared to triangular meshes, meta-meshes are very lightweight and much less compute-demanding. The meta-mesh can also work as a base mesh reusable for conveniently and efficiently triangulating lattice structures with arbitrary resolutions. A CPU+GPU asynchronous meta-meshing pipeline has been developed to efficiently generate meta-meshes from lattice structures. It shifts from the thread-centric GPU algorithm design paradigm commonly used in CAD to the recent warp-centric design paradigm to achieve high performance. This is achieved by a new data compression method, a GPU cache-aware data structure, and a workload-balanced scheduling method that can significantly reduce memory divergence and branch divergence. Experimenting with various billion-scale lattice structures, the proposed method is seen to be two orders of magnitude faster than previously achievable.

This paper investigates the numerical approximation of ground states of rotating Bose-Einstein condensates. This problem requires the minimization of the Gross-Pitaevskii energy $E$ on a Riemannian manifold $\mathbb{S}$. To find a corresponding minimizer $u$, we use a generalized Riemannian gradient method that is based on the concept of Sobolev gradients in combination with an adaptively changing metric on the manifold. By a suitable choice of the metric, global energy dissipation for the arising gradient method can be proved. The energy dissipation property in turn implies global convergence to the density $|u|^2$ of a critical point $u$ of $E$ on $\mathbb{S}$. Furthermore, we present a precise characterization of the local convergence rates in a neighborhood of each ground state $u$ and how these rates depend on the first spectral gap of $E^{\prime\prime}(u)$ restricted to the $L^2$-orthogonal complement of $u$. With this we establish the first convergence results for a Riemannian gradient method to minimize the Gross-Pitaevskii energy functional in a rotating frame. At the same, we refine previous results obtained in the case without rotation. The major complication in our new analysis is the missing isolation of minimizers, which are at most unique up to complex phase shifts. For that, we introduce an auxiliary iteration in the tangent space $T_{\mathrm{i} u} \mathbb{S}$ and apply the Ostrowski theorem to characterize the asymptotic convergence rates through a weighted eigenvalue problem. Afterwards, we link the auxiliary iteration to the original Riemannian gradient method and bound the spectrum of the weighted eigenvalue problem to obtain quantitative convergence rates. Our findings are validated in numerical experiments.

This paper proposes a second-order accurate numerical scheme for the Patlak-Keller-Segel system with various mobilities for the description of chemotaxis. Formulated in a variational structure, the entropy part is novelly discretized by a modified Crank-Nicolson approach so that the solution to the proposed nonlinear scheme corresponds to a minimizer of a convex functional. A careful theoretical analysis reveals that the unique solvability and positivity-preserving property could be theoretically justified. More importantly, such a second order numerical scheme is able to preserve the dissipative property of the original energy functional, instead of a modified one. To the best of our knowledge, the proposed scheme is the first second-order accurate one in literature that could achieve both the numerical positivity and original energy dissipation. In addition, an optimal rate convergence estimate is provided for the proposed scheme, in which rough and refined error estimate techniques have to be included to accomplish such an analysis. Ample numerical results are presented to demonstrate robust performance of the proposed scheme in preserving positivity and original energy dissipation in blowup simulations.