arXiv subjects

Victor M. Calo

Publications and source records attributed to Victor M. Calo.

At least 37 records · Page 2Linked to original sources

arXiv2021

A continuum theory for mineral solid solutions undergoing chemo-mechanical processes

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 metamorphic minerals to showcase and verify the chemo-mechanical interactions of mineral solid solutions that undergo spinodal decomposition, chemical reactions, and deformation.

physics.geo-ph↗

arXiv2021

Extended Larché--Cahn framework for reactive Cahn--Hilliard multicomponent systems

At high temperature and pressure, solid diffusion and chemical reactions between rock minerals lead to phase transformations. Chemical transport during uphill diffusion causes phase separation, that is, spinodal decomposition. Thus, to describe the coarsening kinetics of the exsolution microstructure, we derive a thermodynamically consistent continuum theory for the multicomponent Cahn--Hilliard equations while accounting for multiple chemical reactions and neglecting deformations. Our approach considers multiple balances of microforces augmented by multiple constituent content balance equations within an extended Larché--Cahn framework. As for the Larché--Cahn framework, we incorporate into the theory the Larché--Cahn derivatives with respect to the phase fields and their gradients. We also explain the implications of the resulting constrained gradients of the phase fields in the form of the gradient energy coefficients. Moreover, we derive a configurational balance that includes all the associated configurational fields in agreement with the Larché--Cahn framework. We study phase separation in a three-component system whose microstructural evolution depends upon the reaction-diffusion interactions and to analyze the underlying configurational fields. This simulation portrays the interleaving between the reaction and diffusion processes and how the configurational tractions drive the motion of interfaces.

physics.geo-ph↗

arXiv2020

Goal-oriented adaptivity for a conforming residual minimization method in a dual discontinuous Galerkin norm

We propose a goal-oriented mesh-adaptive algorithm for a finite element method stabilized via residual minimization on dual discontinuous-Galerkin norms. By solving a saddle-point problem, this residual minimization delivers a stable continuous approximation to the solution on each mesh instance and a residual projection onto a broken polynomial space, which is a robust error estimator to minimize the discrete energy norm via automatic mesh refinement. In this work, we propose and analyze a goal-oriented adaptive algorithm for this stable residual minimization. We solve the primal and adjoint problems considering the same saddle-point formulation and different right-hand sides. By solving a third stable problem, we obtain two efficient error estimates to guide goal-oriented adaptivity. We illustrate the performance of this goal-oriented adaptive strategy on advection-diffusion-reaction problems.

math.NA↗

arXiv2020

An automatic-adaptivity stabilized finite element method via residual minimization for heterogeneous, anisotropic advection-diffusion-reaction problems

In this paper, we describe a stable finite element formulation for advection-diffusion-reaction problems that allows for robust automatic adaptive strategies to be easily implemented. We consider locally vanishing, heterogeneous, and anisotropic diffusivities, as well as advection-dominated diffusion problems. The general stabilized finite element framework was described and analyzed in arXiv:1907.12605v3 for linear problems in general, and tested for pure advection problems. The method seeks for the discrete solution through a residual minimization process on a proper stable discontinuous Galerkin (dG) dual norm. This technique leads to a saddle-point problem that delivers a stable discrete solution and a robust error estimate that can drive mesh adaptivity. In this work, we demonstrate the efficiency of the method in extreme scenarios, delivering stable solutions. The quality and performance of the solutions are comparable to classical discontinuous Galerkin formulations in the respective discrete space norm on each mesh. Meanwhile, this technique allows us to solve on coarse meshes and adapt the solution to achieve a user-specified solution quality.

math.NA↗

arXiv2020

A nonlinear weak constraint enforcement method for advection-dominated diffusion problems

We devise a stabilized method to weakly enforce bound constraints in the discrete solution of advection-dominated diffusion problems. This method combines a nonlinear penalty formulation with a discontinuous Galerkin-based residual minimization method. We illustrate the efficiency of this scheme for both uniform and adaptive meshes through proper numerical examples.

math.NA↗

arXiv2020

Incompressible flow modeling using an adaptive stabilized finite element method based on residual minimization

We model incompressible flows with an adaptive stabilized finite element method Stokes flows, which solves a discretely stable saddle-point problem to approximate the velocity-pressure pair. Additionally, this saddle-point problem delivers a robust error estimator to guide mesh adaptivity. We analyze the accuracy of different discrete velocity-pressure pairs of continuous finite element spaces, which do not necessarily satisfy the discrete inf-sup condition. We validate the framework's performance with numerical examples.

math.NA↗

arXiv2020

Refined isogeometric analysis for generalized Hermitian eigenproblems

We use the refined isogeometric analysis (rIGA) to solve generalized Hermitian eigenproblems $({Ku=λMu})$. The rIGA framework conserves the desirable properties of maximum-continuity isogeometric analysis (IGA) discretizations while reducing the computation cost of the solution through partitioning the computational domain by adding zero-continuity basis functions. As a result, rIGA enriches the approximation space and decreases the interconnection between degrees of freedom. We compare computational costs of rIGA versus those of IGA when employing a Lanczos eigensolver with a shift-and-invert spectral transformation. When all eigenpairs within a given interval ${[λ_s,λ_e]}$ are of interest, we select several shifts ${σ_k\in[λ_s,λ_e]}$ using a spectrum slicing technique. For each shift $σ_k$, the cost of factorization of the spectral transformation matrix ${K-σ_k M}$ drives the total computational cost of the eigensolution. Several multiplications of the operator matrices ${(K-σ_k M)^{-1} M}$ by vectors follow this factorization. Let $p$ be the polynomial degree of basis functions and assume that IGA has maximum continuity of ${p-1}$, while rIGA introduces $C^0$ separators to minimize the factorization cost. For this setup, our theoretical estimates predict computational savings to compute a fixed number of eigenpairs of up to ${O(p^2)}$ in the asymptotic regime, that is, large problem sizes. Yet, our numerical tests show that for moderately-sized eigenproblems, the total computational cost reduction is $O(p)$. Nevertheless, rIGA improves the accuracy of every eigenpair of the first $N_0$ eigenvalues and eigenfunctions. Here, we allow $N_0$ to be as large as the total number of eigenmodes of the original maximum-continuity IGA discretization.

math.NA↗

arXiv2020

Adaptive stabilized finite elements: Continuation analysis of compaction banding in geomaterials

Under compressive creep, visco-plastic solids experiencing internal mass transfer processes have been recently proposed to accommodate singular cnoidal wave solutions, as material instabilities at the stationary wave limit. These instabilities appear when the loading rate is significantly faster than the capability of the material to diffuse internal perturbations and lead to localized failure features (e.g., cracks and compaction bands). This type of solution, generally found in fluids, has strong nonlinearities and periodic patterns. Due to the singular nature of the solutions, the applicability of the theory is currently limited. Additionally, effective numerical tools require proper regularization to overcome the challenges that singularity induces. We focus on the numerical treatment of the governing equation using a nonlinear approach building on a recent adaptive stabilized finite element method. This method provides a residual representation to drive adaptive mesh refinement, a particularly useful feature for the problem at hand. We compare against analytical and standard finite element solutions to demonstrate the performance of our approach. We then investigate the sensitivity of the diffusivity ratio, main parameter of the problem, and identify multiple possible solutions, with multiple stress peaks. Finally, we show the evolution of the spacing between peaks for all solutions as a function of that parameter.

physics.comp-ph↗

arXiv2019

An adaptive stabilized conforming finite element method via residual minimization on dual discontinuous Galerkin norms

We design and analyze a new adaptive stabilized finite element method. We construct a discrete approximation of the solution in a continuous trial space by minimizing the residual measured in a dual norm of a discontinuous test space that has inf-sup stability. We formulate this residual minimization as a stable saddle-point problem which delivers a stabilized discrete solution and a residual representation that drives the adaptive mesh refinement. Numerical results on an advection-reaction model problem show competitive error reduction rates when compared to discontinuous Galerkin methods on uniformly refined meshes and smooth solutions. Moreover, the technique leads to optimal decay rates for adaptive mesh refinement and solutions having sharp layers.

math.NA↗

arXiv2019

Split generalized-$α$ method: A linear-cost solver for a modified generalized-method for multi-dimensional second-order hyperbolic systems

We propose a variational splitting technique for the generalized-$α$ method to solve hyperbolic partial differential equations. We use tensor-product meshes to develop the splitting method, which has a computational cost that grows linearly with respect to the total number of degrees of freedom for multi-dimensional problems. We consider standard $C^0$ finite elements as well as smoother B-splines in isogeometric analysis for the spatial discretization. We also study the spectrum of the amplification matrix to establish the unconditional stability of the method. We then show that the stability behavior affects the overall behavior of the integrator on the entire interval and not only at the limits $0$ and $\infty$. We use various examples to demonstrate the performance of the method and the optimal approximation accuracy. For the numerical tests, we compute the $L_2$ and $H^1$ norms to show the optimal convergence of the discrete method in space and second-order accuracy in time.

math.NA↗

arXiv2019

Higher-order generalized-$α$ methods for hyperbolic problems

The generalized-$α$ time-marching method provides second-order accuracy in time and controls the numerical dissipation in the high-frequency region of the discrete spectrum. This method includes a wide range of time integrators. We increase the order of accuracy of the method while keeping the unconditional stability and the user-control on the high-frequency numerical dissipation. The dissipation is controlled by a single parameter as in the original method. Our high-order schemes require simple modifications of the available implementations of the generalized-$α$ method.

math.NA↗

arXiv2019

Automatic Variationally Stable Analysis for FE Computations: An Introduction

We introduce an automatic variationally stable analysis (AVS) for finite element (FE) computations of scalar-valued convection-diffusion equations with non-constant and highly oscillatory coefficients. In the spirit of least squares FE methods, the AVS-FE method recasts the governing second order partial differential equation (PDE) into a system of first-order PDEs. However, in the subsequent derivation of the equivalent weak formulation, a Petrov-Galerkin technique is applied by using different regularities for the trial and test function spaces. We use standard FE approximation spaces for the trial spaces, which are C0, and broken Hilbert spaces for the test functions. Thus, we seek to compute pointwise continuous solutions for both the primal variable and its flux (as in least squares FE methods), while the test functions are piecewise discontinuous. To ensure the numerical stability of the subsequent FE discretizations, we apply the philosophy of the discontinuous Petrov-Galerkin (DPG) method by Demkowicz and Gopalakrishnan, by invoking test functions that lead to unconditionally stable numerical systems (if the kernel of the underlying differential operator is trivial). In the AVS-FE method, the discontinuous test functions are ascertained per the DPG approach from local, decoupled, and well-posed variational problems, which lead to best approximation properties in terms of the energy norm. We present various 2D numerical verifications, including convection-diffusion problems with highly oscillatory coefficients and extremely high Peclet numbers, up to a billion. These show the unconditional stability without the need for any upwind schemes nor any other artificial numerical stabilization. The results are not highly diffused for convection-dominated problems ...

math.NA↗

arXiv2019

High-order generalized-$α$ methods

The generalized-$α$ method encompasses a wide range of time integrators. The method possesses high-frequency dissipation while minimizing unwanted low-frequency dissipation and the numerical dissipation can be controlled by the user. The method is unconditionally stable and is of second-order accuracy in time. We extend the second-order generalized-$α$ method to third-order in time while the numerical dissipation can be controlled in a similar fashion. We establish that the third-order method is unconditionally stable. We discuss a possible path to the generalization to higher order schemes. All these high-order schemes can be easily implemented into programs that already contain the second-order generalized-$α$ method.

math.NA↗

arXiv2019

Residual minimization for isogeometric analysis in reduced and mixed forms

Most variational forms of isogeometric analysis use highly-continuous basis functions for both trial and test spaces. For a partial differential equation with a smooth solution, isogeometric analysis with highly-continuous basis functions for trial space results in excellent discrete approximations of the solution. However, we observe that high continuity for test spaces is not necessary. In this work, we present a framework which uses highly-continuous B-splines for the trial spaces and basis functions with minimal regularity and possibly lower order polynomials for the test spaces. To realize this goal, we adopt the residual minimization methodology. We pose the problem in a mixed formulation, which results in a system governing both the solution and a Riesz representation of the residual. We present various variational formulations which are variationally-stable and verify their equivalence numerically via numerical tests.

math.NA↗

arXiv2018

A variationally separable splitting for the generalized-$α$ method for parabolic equations

We present a variationally separable splitting technique for the generalized-$α$ method for solving parabolic partial differential equations. We develop a technique for a tensor-product mesh which results in a solver with a linear cost with respect to the total number of degrees of freedom in the system for multi-dimensional problems. We consider finite elements and isogeometric analysis for the spatial discretization. The overall method maintains user-controlled high-frequency dissipation while minimizing unwanted low-frequency dissipation. The method has second-order accuracy in time and optimal rates ($h^{p+1}$ in $L^2$ norm and $h^p$ in $L^2$ norm of $\nabla u$) in space. We present the spectrum analysis on the amplification matrix to establish that the method is unconditionally stable. Various numerical examples illustrate the performance of the overall methodology and show the optimal approximation accuracy.

math.NA↗

arXiv2018

Variational Formulations for Explicit Runge-Kutta Methods

Variational space-time formulations for Partial Differential Equations have been of great interest in the last decades. While it is known that implicit time marching schemes have variational structure, the Galerkin formulation of explicit methods in time remains elusive. In this work, we prove that the explicit Runge-Kutta methods can be expressed as discontinuous Petrov-Galerkin methods both in space and time. We build trial and test spaces for the linear diffusion equation that lead to one, two, and general stage explicit Runge-Kutta methods. This approach enables us to design explicit time-domain (goal-oriented) adaptive algorithms

math.NA↗

arXiv2015

Localized Harmonic Characteristic Basis Functions for Multiscale Finite Element Methods

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 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 expansions 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 that generate the multiscale finite element space where the approximation of the solution is computed. Numerical experiments show the good performance of the proposed methodology.

math.NA↗

arXiv2015

Multiscale stabilization for convection-dominated diffusion in heterogeneous media

We develop a Petrov-Galerkin stabilization method for multiscale convection-diffusion transport systems. Existing stabilization techniques add a limited number of degrees of freedom in the form of bubble functions or a modified diffusion, which may not sufficient to stabilize multiscale systems. We seek a local reduced-order model for this kind of multiscale transport problems and thus, develop a systematic approach for finding reduced-order approximations of the solution. We start from a Petrov-Galerkin framework using optimal weighting functions. We introduce an auxiliary variable to a mixed formulation of the problem. The auxiliary variable stands for the optimal weighting function. The problem reduces to finding a test space (a reduced dimensional space for this auxiliary variable), which guarantees that the error in the primal variable (representing the solution) is close to the projection error of the full solution on the reduced dimensional space that approximates the solution. To find the test space, we reformulate some recent mixed Generalized Multiscale Finite Element Methods. We introduce snapshots and local spectral problems that appropriately define local weight and trial spaces. In particular, we use energy minimizing snapshots and local spectral decompositions in the natural norm associated with the auxiliary variable. The resulting spectral decomposition adaptively identifies and builds the optimal multiscale space to stabilize the system. We discuss the stability and its relation to the approximation property of the test space. We design online basis functions, which accelerate convergence in the test space, and consequently, improve stability. We present several numerical examples and show that one needs a few test functions to achieve an error similar to the projection error in the primal variable irrespective of the Peclet number.

math.NA↗