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Zebra rocks, characterized by their striking reddish‐brown stripes, rods, and spots of hematite (Fe‐oxide), showcase complex self‐organized patterns formed under far‐from‐equilibrium conditions. Despite their ease of recognition, the underlying mechanisms of pattern‐forming processes remain elusive. We introduce a novel advection‐dominated phase‐field model that effectively replicates the Liesegang‐like patterns observed in Zebra rocks. This numerical model leverages the concept of phase separation, a well‐established principle governing Liesegang phenomena in a two‐dimensional setting. Our findings reveal that initial solute concentration and fluid flow velocity are critical determinants in pattern morphologies. We quantitatively explain the spacing and width of a specific Liesegang‐like pattern category. Furthermore, the model demonstrates that vanishingly low initial concentrations promote the formation of oblique patterns, with inclination angles influenced by rock heterogeneity. Additionally, we establish a quantitative relationship between band thickness and geological parameters for orthogonal bands. This enables the characterization of critical geological parameters based solely on static patterns observed in Zebra rocks, providing valuable insights into their formation environments. The diverse patterns in Zebra rocks share similarities with morphologies observed on early Earth and Mars, such as banded iron formations and hematite spherules. Our model, therefore, offers a plausible explanation for the formation mechanisms of these patterns and presents a powerful tool for deciphering the geochemical environments of their origin. Zebra rocks, known for their unique red and brown stripes and spots, hold clues to how similar patterns formed on early Earth and Mars. We have developed a new model to explain how these intriguing patterns form. The model suggests that the flow of fluids and the initial amount and location of dissolved iron‐oxide (rust) in water plays a big role in shaping the final Zebra rock design. The speed of the flow and the initial amount of rust can create different stripe and spot patterns, just like the ones in Zebra rocks. By studying these rocks, we can potentially decipher fluid flow scenarios of ancient environments on both Earth and Mars. Patterns enable identification of flow velocity and concentration when observedFive Liesegang‐like patterns are replicated by varying flow velocities and initial concentrationsOne class of inclined patterns is identified in a very narrow flow velocity and low concentration regime Patterns enable identification of flow velocity and concentration when observed Five Liesegang‐like patterns are replicated by varying flow velocities and initial concentrations One class of inclined patterns is identified in a very narrow flow velocity and low concentration regime

The direct numerical simulation of metal additive manufacturing processes such as laser powder bed fusion is challenging due to the vast differences in spatial and temporal scales. Classical approaches based on locally refined finite elements combined with time-stepping schemes can only address the spatial multi-scale nature and provide only limited scaling potential for massively parallel computations. We address these shortcomings in a space-time Galerkin framework where the finite element interpolation also includes the temporal dimension. In this setting, we construct four-dimensional meshes that are locally refined towards the laser spot and allow for varying temporal accuracy depending on the position in space. By splitting the mesh into conforming time-slabs, we recover a stepwise solution to solve the space-time problem locally in time at this slab; additionally, we can choose time-slab sizes significantly larger than classical time-stepping schemes. As a result, we believe this setting to be well suited for large-scale parallelization. In our work, we use a continuous Galerkin–Petrov formulation of the nonlinear heat equation with an apparent heat capacity model to account for the phase change. We validate our approach by computing the AMB2018-02 benchmark, where we obtain an excellent agreement with the measured melt pool shape. Using the same setup, we demonstrate the performance potential of our approach by hatching a square area with a laser path length of about one meter.

Mineral precipitation can form complex patterns under non‐equilibrium conditions, in which two representative patterns are rhythmic Liesegang stripes and fractal dendrites. Interestingly, both patterns occur in the same rock formations, including various dendritic morphologies found in different rocks, such as limestone and sandstone. However, the underlying mechanism for selecting the vastly different mineral precipitation patterns remains unclear. We use a phase‐field model to reveal the mechanisms driving pattern selection in mineral precipitation. Simulations allow us to explore the effects of diffusion parameters on determining the dendritic morphologies. We also propose a general criterion to distinguish the resulting dendrites in simulations and field observations based on a qualitative visual distinction into three categories and a quantitative fractal dimension (FD) phase diagram. Using this model, we reproduce the classified dendrites in the field and invert for the key parameters that reflect the intrinsic material properties and geological environments. This study provides a quantitative tool for identifying the morphology selection mechanism with potential applications to geological field studies, exploration for resource evaluation, and other potential industrial applications. Dendrites are branched, tree‐like structures that are found in various natural and biological systems. In the context of mineral precipitation or crystal growth, dendrites refer to the branching patterns formed by the deposition of minerals or crystals in a non‐equilibrium environment. Other patterns frequently found in these reactive systems are concentric ring‐like or striped patterns called Liesegang patterns. Although these patterns occur in the same rock formations, the mechanism for selecting different mineral precipitation patterns is not well understood. Here we use a phase‐field model to explore how different diffusion parameters affect the selection of dendritic morphologies. We propose a criterion to distinguish different dendrites in simulations and field observations based on a qualitative visual distinction (needle‐like, tree‐like and seaweed‐like) and a quantitative fractal dimension phase diagram. The model can be deployed to invert for the intrinsic rock properties as well as the geological environments these dendrites grow in. The numerical scheme is robust and efficient, allowing easy implementation of add‐on features in the future for potential applications across scientific and engineering disciplines. Fractal dendritic morphology diagram reproduces natural dendrite patternsCompared to Liesegang patterns, dendrites require higher fluid diffusivity and tend to grow along interfacesInversion of phase‐field diffusivity and self‐diffusion coefficient of solute is possible from photographic images of dendritic patterns Fractal dendritic morphology diagram reproduces natural dendrite patterns Compared to Liesegang patterns, dendrites require higher fluid diffusivity and tend to grow along interfaces Inversion of phase‐field diffusivity and self‐diffusion coefficient of solute is possible from photographic images of dendritic patterns

We solve elliptic systems of equations posed on highly heterogeneous materials. Examples of this class of problems are composite structures and geological processes. We focus on a model problem which is a second-order elliptic equation with discontinuous coefficients. These coefficients represent the conductivity of a composite material. We assume a background with a low conductivity that contains inclusions with different thermal properties. Under this scenario, we design a multiscale finite element method to efficiently approximate solutions. The method is based on an asymptotic expansion of the solution in terms of the ratio between the conductivities. The resulting method constructs (locally) finite element basis functions (one for each inclusion). These bases generate the multiscale finite element space where the approximation of the solution is computed. Numerical experiments show the good performance of the proposed methodology.

This paper presents a phase-field based numerical study on the 3D formation of dendrites due to electrodeposition in an experimental-scale lithium metal battery. Small-scale 3D simulations were firstly conducted to elucidate the characteristics and resolution requirements of the numerical framework. Using a four-fold anisotropy model to simulate the growth of lithium deposition, the dependency of dendrite morphology on charging conditions (ϕb=−0.7Vand ϕb=−1.4V) on a (larger) experimental-scale metal anode was demonstrated. The dendrite shape was found to shift from a smoother, tree-like formation at the lower applied voltage, to a more spike-like, highly branched structure at the higher voltage. The resulting morphological parameters, such as dendrite propagation rates, volume-specific area, and side branching rates, were compared against published experimental data and found to be comparable to the reported ranges for the electrodeposition of spike- or tree-like metal dendrites. This finding supports our previous observation that dendrite formation is connected to the competition between the lithium cation diffusion and electric migration, generating an uneven distribution of Li+on the electrode surface. This observation also gives insight into dendrite inhibition strategies focusing on enhancing the diffusion of lithium ions to achieve a more uniform concentration field on the anode surface.

This work applies machine learning to holistically interrogate the influence of metallurgical factors, such as chemical composition, heat treatment, and mechanical properties, on the stress corrosion cracking resistance of corrosion-resistant alloys. Particularly, we explored the effect of nickel in reducing the stress corrosion cracking susceptibility in boiling magnesium chloride, arguably a controversial topic since Copson’s 1959 seminal publication. This paper offers insights into the synergies of nickel with other alloying elements that ultimately impact the resistance to stress corrosion cracking. Furthermore, a more detailed description of statistical patterns in the so-called Copson curve is provided.

Self‐organizing diffusion‐reaction systems naturally form complex patterns under far‐from‐equilibrium conditions. A representative example is the rhythmic concentration pattern of Fe‐oxides in Zebra rocks; these patterns include reddish‐brown stripes, rounded rods, and elliptical spots. Similar patterns are observed in the banded iron formations, which are presumed to have formed in the early earth under global glaciation. We propose that such patterns can be used directly (e.g., by computer‐vision‐analysis) to infer basic quantities relevant to their formation giving information on generalized chemical gradients. Here we present a phase‐field model that quantitatively captures the distinct Zebra rock patterns based on the concept of phase separation that describes the process forming Liesegang stripes. We find that diffusion coefficients (the bulk self‐diffusivities of the native species and the mobility of the reaction product) play an essential role in controlling the appearance of regular stripe patterns as well as the transition from stripes to spots. The numerical results are carefully benchmarked with well‐established empirical spacing law, width law, timing law and the Matalon‐Packter law. Using this model, we invert for the important process parameters originating from the intrinsic material properties, the self‐diffusivity ratio and the mobility of Fe‐oxides with a series of Zebra rock samples. This study allows a quantitative prediction of the generalized chemical gradients in mineralized source rocks without intrusive measurements, providing a better intuition for the mineral exploration space. Patterns in nature are observed in disparate fields of science in biology, geology, mechanics, atmospheric physics, chemistry and others. A unifying principle to decipher those patterns is using a reaction‐diffusion approach, as employed here, when implemented in numerical simulations, can deliver a perfect match to patterns observed in nature. Here we go one step further and show that dynamic coefficients describing the concentrations and mobility of valuable species such as iron oxide can be derived from the images. The proposed algorithms are benchmarked on the Zebra banded rock in Western Australia. The broader impact of the presented work includes the development of a future exploration tool based on computer‐version, revealing high grades of iron in the prominent worldwide banded iron formations which bear similar characteristic stripes. Using a physics‐based model, our formulation also captures the interesting phenomenon of transition from bands to spots that has not been addressed before. Inversion of Cahn‐Hilliard dynamic coefficients from photographic images of Zebra rock reveals multiphysics coupled processesTransition from stripes to spots is triggered by low reaction product (here Fe‐Oxyhydroxite) mobilityApplication of computer‐vision algorithms in conjunction with inversion may allow novel geophysical exploration tools for ores Inversion of Cahn‐Hilliard dynamic coefficients from photographic images of Zebra rock reveals multiphysics coupled processes Transition from stripes to spots is triggered by low reaction product (here Fe‐Oxyhydroxite) mobility Application of computer‐vision algorithms in conjunction with inversion may allow novel geophysical exploration tools for ores

We consider a class of two- and three-dimensional h-refined meshes generated by an adaptive finite element method. We introduce an element partition tree, which controls the execution of the multi-frontal solver algorithm over these refined grids. We propose and study algorithms with polynomial computational cost for the optimization of these element partition trees. The trees provide an ordering for the elimination of unknowns. The algorithms automatically optimize the element partition trees using extensions of dynamic programming. The construction of the trees by the dynamic programming approach is expensive. These generated trees cannot be used in practice, but rather utilized as a learning tool to propose fast heuristic algorithms. In this first part of our paper we focus on the dynamic programming approach, and draw a sketch of the heuristic algorithm. The second part will be devoted to a more detailed analysis of the heuristic algorithm extended for the case of hp-adaptive grids.

Explicit time-marching schemes are popular for solving time-dependent partial differential equations; one of the biggest challenges these methods suffer is increasing the critical time-marching step size that guarantees numerical stability. In general, there are two ways to increase the critical step size. One is to reduce the stiffness of the spatially discretized system, while the other is to design time-marching schemes with larger stability regions. In this paper, we focus on the recently proposed explicit generalized- α method for second-order hyperbolic equations and increase the critical step size by reducing the stiffness of the isogeometric-discretized system. In particular, we apply boundary penalization to lessen the system's stiffness. For p th order C p − 1 isogeometric elements, we show numerically that the critical step size increases by a factor of p 2 − 3 p + 6 4 , which indicates the advantages of using the proposed method, especially for high-order elements. Various examples in one and two dimensions validate the performance of the proposed technique. • The second-order hyperbolic equation is solved by the boundary-penalized isogeometric analysis in space. • An explicit generalized- α method is applied for the temporal1 discretization. • Optimal numerical accuracy is guaranteed both in space and in time. • The critical time-marching step size is increased by the use of the boundary penalization technique. [ABSTRACT FROM AUTHOR]

We describe a phase-field model for the electrodeposition process that forms dendrites within metal-anode batteries. We derive the free energy functional model, arriving at a system of partial differential equations that describe the evolution of a phase field, the lithium-ion concentration, and an electric potential. We formulate, discretize, and solve the set of partial differential equations describing the coupled electrochemical interactions during a battery charge cycle using an open-source finite element library. The open-source library allows us to use parallel solvers and time-marching adaptivity. We describe two- and three-dimensional simulations; these simulations agree with experimentally-observed dendrite growth rates and morphologies reported in the literature.

The main result of this paper is the generalization of the concept of variational splitting proposed by Behnoudfar et al. (2018) into higher-order time-stepping schemes. The key novelty of the technique proposed here is the following: when using the isogeometric finite elements, one can factorize the matrix of the linear system and obtain the linear complexity. This matrix has the form M + η K , where M is the mass matrix, K is the stiffness matrix. Following the technique of Behnoudfar et al. (2018), and factorizing M + η K one would get the additional error of order η 2. We propose the technique to couple the possibility of factorization of M + η K with the higher-order time-stepping schemes and preserve this higher order. Thus, we introduce a variational splitting for implicit, linear multistep methods of order s for parabolic partial differential equations. Our technique exploits the tensor-product structure of the discretization in space (mesh and basis functions) to derive an update strategy with a linear cost regarding the total number of degrees of freedom for multi-dimensional problems. As an example, we consider two popular methods: Adams–Moulton (AM) and backward differentiation formulae (BDF). We derive the corresponding coefficients for AM and BDF methods. We present a C++ implementation of the time-integration schemes, targeting shared-memory Linux clusters. We present numerical results for the heat transfer problem and advection–diffusion problem. We also discuss the limitations of the method and possible extensions overcoming this limitation by using our method as a preconditioner for an iterative solver. • We present the new time integration schemes suitable for direction-splitting. • We present the spectral analysis of the stability region and CFL of the new method. • We show numerical accuracy, linear computational cost, and high order of convergence. • Our solver use higher-order and continuity base functions on tensor product grids. • We provide a C++ library implementing a linear cost Adams–Moulton and BDF schemes. [ABSTRACT FROM AUTHOR]

We study the problem of finding stretch-minimising stream surfaces in a divergence-free vector field. These surfaces are generated by motions of seed curves that propagate through the field in a stretch minimising manner, i.e., they move without stretching or shrinking, preserving the length of their arbitrary arc. In general fields, such curves may not exist. However, the divergence-free constraint gives rise to these ‘stretch-free’ curves that are locally arc-length preserving when infinitesimally propagated. Several families of stretch-free curves are identified and used as initial guesses for stream surface generation. These surfaces are subsequently globally optimised to obtain the best stretch-minimising stream surfaces in a given divergence-free vector field. Our algorithm was tested on benchmark datasets, proving its applicability to incompressible fluid flow simulations, where our stretch-minimising stream surfaces realistically reflect the flow of a flexible univariate object. [ABSTRACT FROM AUTHOR]

This paper presents a detailed description of phase-field models of electrodeposition in lithium-anode batteries, along with underlying assumptions and parameters commonly employed. We simulate the coupled electrochemical interactions during a battery charge cycle using finite elements on open-source packages, allowing for parallel computation and time step adaptivity. We compare conventional free energy and grand canonical formulations. We obtain agreement between 1D phase-field simulations and the theoretical Faradic reaction kinetics. We study the mesh-induced errors through spatial convergence analysis. These simulations results set the groundwork for 2D and 3D simulations of dendritic metal electrodeposition in batteries.

Abstract: We revisit the finite element analysis of convection-dominated flow problems within the recently developed Discontinuous Petrov–Galerkin (DPG) variational framework. We demonstrate how test function spaces that guarantee numerical stability can be computed automatically with respect to the optimal test space norm. This makes the DPG method not only stable but also robust, that is, uniformly stable with respect to the Péclet number in the current application. We employ discontinuous piecewise Bernstein polynomials as trial functions and construct a subgrid discretization that accounts for the singular perturbation character of the problem to resolve the corresponding optimal test functions. We also show that a smooth B-spline basis has certain computational advantages in the subgrid discretization. The overall effectiveness of the algorithm is demonstrated on two problems for the linear advection–diffusion equation. [Copyright &y& Elsevier]

Abstract: We discuss the use of time adaptivity applied to the one dimensional diffusive wave approximation to the shallow water equations. A simple and computationally economical error estimator is discussed which enables time-step size adaptivity. This robust adaptive time discretization corrects the initial time step size to achieve a user specified bound on the discretization error and allows time step size variations of several orders of magnitude. In particular, the one dimensional results presented in this work feature a change of four orders of magnitudes for the time step over the entire simulation. [Copyright &y& Elsevier]

In this paper we describe the history, performance, and design concepts of direct solvers for algebraic systems resulting from Galerkin discretizations of partial differential equations. Popular direct solver implementations of Gaussian elimination (also known as LU factorization) are introduced and briefly analyzed. We discuss three of the most relevant aspects influencing the performance of direct solvers on this kind of algebraic systems. First, the ordering of the degrees of freedom of the algebraic system has a significant impact on the solver performance, solution speed and memory requirements. The impact of unknowns ordering for elimination is exemplified and alternative ordering algorithms are described and compared. Second, the effect of round-off error on the simulation results is discussed. We detail this effect for uniform grids where the impact of round-off error on the solution is controlled by the condition number of the matrix in terms of the element size, but is independent of the polynomial order of approximation. Additionally, we discuss the link between unknown ordering and round-off error. Third, we describe the impact of the connectivity pattern (graph) of the basis functions on the performance of direct solvers. Variations in the connectivity structure of the resulting discrete system have severe impact on performance of the solver. That is, the resources needed to factorize the system strongly depend on its connectivity graph. Less connected graphs are cheaper to solve, that is, C0finite element discretizations are cheaper to solve with direct solvers than Cp-1discretizations.

• We consider residual minimization method with discontinuous Galerkin formulation. • We focus on two-dimensional Stokes problem with manufactured and cavity flow problems. • We discretize with B-spline basis functions from isogeometric analysis. • We consider different trial and test spaces to test convergence of iterative and direct solvers. • We show that our method can be extend to three-dimensional problems. In this paper, we introduce a stable isogeometric analysis discretization of the Stokes system of equations. We use this standard constrained problem to demonstrate the flexibility and robustness of the residual minimization method on dual stable norms [16] , which unlocks the extraordinary approximation power of isogeometric analysis [44]. That is, we introduce an isogeometric residual minimization method (IRM) for the Stokes equations, which minimizes the residual in a dual discontinuous Galerkin norm; thus we use the acronym DGiRM. Following Calo et al. [16] , we start from an inf-sup stable discontinuous Galerkin (DG) formulation to approximate in a highly continuous trial space that minimizes the dual norm of the residual in a discontinuous test space. We demonstrate the performance and robustness of the methodology considering a manufactured solution and the well-known lid-driven cavity flow problem. First, we use a multi-frontal direct solver, and, using the Pareto front, compare the resulting numerical accuracy and the computational cost expressed by the number of floating-point operations performed by the direct solver algorithm. Second, we use an iterative solver. We measure the number of iterations required when increasing the mesh size and how the configuration of spaces affect the resulting accuracy. This paper is an extension of the paper A Stable Discontinuous Galerkin Based Isogeometric Residual Minimization for the Stokes Problem by M. Łoś, et al. (2020) published in Lecture Notes in Computer Science. In this extended version, we deepen in the mathematical aspects of the DGiRM, include the iterative solver algorithm and implementation, and discuss the influence of different discretization spaces on the iterative solver's convergence. [ABSTRACT FROM AUTHOR]

• Optimally-blended quadratures for isogeometric analysis are applied for spectral approximation of various operators. • Dispersion error is minimized for diffusion-reaction operators with both Dirichlet and Neumann boundary conditions. • Various numerical examples in 1D and 3D demonstrate the performance of the techniques. We study the spectral approximation of a second-order elliptic differential eigenvalue problem that arises from structural vibration problems using isogeometric analysis. In this paper, we generalize recent work in this direction. We present optimally-blended quadrature rules for the isogeometric spectral approximation of a diffusion-reaction operator with both Dirichlet and Neumann boundary conditions. The blended rules improve the accuracy of the isogeometric approximation. In particular, the optimal blending rules minimize the dispersion error and lead to two extra orders of super-convergence in the eigenvalue error. Various numerical examples (including the Schrödinger operator for quantum mechanics) in one and three spatial dimensions demonstrate the performance of the blended rules. [ABSTRACT FROM AUTHOR]

We use a potential flow solver to investigate the aerodynamic aspects of flapping flights in enclosed spaces. The enclosure effects are simulated by the method of images. Our study complements previous aerodynamic analyses which considered only the near-ground flight. The present results show that flying in the proximity of an enclosure affects the aerodynamic performance of flapping wings in terms of lift and thrust generation and power consumption. It leads to higher flight efficiency and more than 5% increase of the generation of lift and thrust.