We describe the development of a high-performance solution framework for isogeometric discrete differential forms based on B-splines: PetIGA-MF. Built on top of PetIGA, PetIGA-MF is a general multi-field discretization tool. To test the capabilities of our implementation, we solve different viscous flow problems such as Darcy, Stokes, Brinkman, and Navier-Stokes equations. Several convergence benchmarks based on manufactured solutions are presented assuring optimal convergence rates of the approximations, showing the accuracy and robustness of our solver.

We develop the energy budget equation of the coupled Navier-Stokes-Cahn-Hilliard (NSCH) system. We use the NSCH equations to model the dynamics of liquid droplets in a liquid continuum. Buoyancy effects are accounted for through the Boussinesq assumption. We physically interpret each quantity involved in the energy exchange to further insight into the model. Highly resolved simulations involving density-driven flows and merging of droplets allow us to analyze these energy budgets. In particular, we focus on the energy exchanges when droplets merge, and describe flow features relevant to this phenomenon. By comparing our numerical simulations to analytical predictions and experimental results available in the literature, we conclude that modeling droplet dynamics within the framework of NSCH equations is a sensible approach worth further research.

We present the microbalance including the microforces, the first-and second-order microstresses for the Swift-Hohenberg equation concomitantly with their constitutive equations, which are consistent with the free-energy imbalance. We provide an explicit form for the microstress structure for a free-energy functional endowed with second-order spatial derivatives. Additionally, we generalize the Swift-Hohenberg theory via a proper constitutive process. Finally, we present one highly-resolved three-dimensional numerical simulation to demonstrate the particular form of the resulting microstresses and their interactions in the evolution of the Swift-Hohenberg equation.

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In this work, we present a novel isogeometric analysis discretization for the Navier-StokesCahn-Hilliard equation, which uses divergence-conforming spaces. Basis functions generated with this method can have higher-order continuity, and allow to directly discretize the higherorder operators present in the equation. The discretization is implemented in PetIGA-MF, a high-performance framework for discrete differential forms. We present solutions in a twodimensional annulus, and model spinodal decomposition under shear flow.

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractThe recently introduced divergence-conforming B-spline discretizations allow the construction of smooth discrete velocity-pressure pairs for viscous incompressible flows that are at the same time inf − sup stable and pointwise divergence-free. When applied to the discretized Stokes problem, these spaces generate a symmetric and indefinite saddle-point linear system. The iterative method of choice to solve such system is the Generalized Minimum Residual Method. This method lacks robustness, and one remedy is to use preconditioners. For linear systems of saddle-point type, a large family of preconditioners can be obtained by using a block factorization of the system. In this paper, we show how the nesting of "black-box" solvers and preconditioners can be put together in a block triangular strategy to build a scalable block preconditioner for the Stokes system discretized by divergence-conforming B-splines. Besides the known cavity flow problem, we used for benchmark flows defined on complex geometries: an eccentric annulus and hollow torus of an eccentric annular cross-section.

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Highlights • We present a second-order energy-stable time-integration method for the Swift-Hohenberg equation that suppresses numerical instabilities. • Based on the generalized-α method, numerical dissipation is controlled via the spectral radius. • A detailed energy-stability proof and an estimate for the stabilization parameter are provided. • The proposed stabilization vanishes for sufficiently small time step sizes, recovering the original weak form of the equation. • Numerical results for a pattern-formation example are used to test the convergence and energy-stability. • We compare the primal and mixed formulation in terms of wall-clock time and approximation error.

Recent studies on metamorphic petrology as well as microstructural observations suggest the influence of mechanical effects upon chemically active metamorphic minerals. Thus, the understanding of such a coupling is crucial to describe the dynamics of geomaterials. In this effort, we derive a thermodynamically consistent framework to characterize the evolution of chemically active minerals. We model the metamorphic mineral assemblages as a solid-species solution where the species mass transport and chemical reaction drive the stress generation process. The theoretical foundations of the framework rely on modern continuum mechanics, thermodynamics far from equilibrium, and the phase-field model. We treat the mineral solid solution as a continuum body, and following the Larché and Cahn network model, we define displacement and strain fields. Consequently, we obtain a set of coupled chemo-mechanical equations. We use the aforementioned framework to study single minerals as solid solutions during metamorphism. Furthermore, we emphasise the use of the phase-field framework as a promising tool to model complex multi-physics processes in geoscience. Without loss of generality, we use common physical and chemical parameters found in the geoscience literature to portrait a comprehensive view of the underlying physics. Thereby, we carry out 2D and 3D numerical simulations using material parameters for mineral solid solutions to showcase and verify the chemo-mechanical interactions of mineral solid solutions that undergo spinodal decomposition, chemical reactions, and deformation.