This study introduces a multiscale simulation framework, termed Projection-based Embedded Discrete Fracture Modeling with Algebraic Dynamic Multilevel method (pEDFM-ADM), which integrates an embedded discrete fracture network representation with a fully algebraic, front-tracking-based mesh adaptation strategy. Incorporating a fully implicit scheme, compositional thermodynamics, and algebraic multilevel operators, the framework captures essential subsurface processes such as buoyancy-driven migration, convective dissolution, phase partitioning, and fracture-matrix interactions under geologically realistic conditions. The method constructs a hierarchy of multilevel grids and localized multiscale basis functions that introduce fine-scale heterogeneities at each coarse level. Adaptive mesh refinement and coarsening are driven by local variations in CO₂ mass fraction and executed through algebraic prolongation and restriction operators, enabling efficient projection between grid levels. The framework is systematically evaluated across a sequence of test cases with increasing complexity, including systems with low-permeability flow barriers, highly conductive fractures, striking a trade-off between computational resource and detailed simulation accuracy. Overall, the pEDFM-ADM framework provides a scalable, fully algebraic, and physically adaptive modeling tool for large-scale CO₂ storage simulations in fractured porous media, supporting predictive simulation and risk assessment for long-term carbon sequestration.

Modeling open-hole failure of composites is a complex task, consisting of a highly nonlinear response with interacting failure modes. Numerical modeling of this phenomenon has traditionally been based on the finite element method, but requires to tradeoff between high fidelity and computational cost. To mitigate this shortcoming, recent work has leveraged machine learning to predict the strength of open-hole composite specimens. Here, we also propose using data-based models to tackle open-hole composite failure from a classification point of view. More specifically, we show how to train surrogate models to learn the ultimate failure envelope of an open-hole composite plate under in-plane loading. To achieve this, we solve the classification problem via support vector machine (SVM) and test different classifiers by changing the SVM kernel function. The flexibility of kernel-based SVM also allows us to integrate the recently developed quantum kernels in our algorithm and compare them with the standard radial basis function kernel. Finally, thanks to kernel-target alignment optimization, we tune the free parameters of all kernels to best separate safe and failure-inducing loading states. The results show classification accuracies higher than 90% for RBF, especially after alignment, followed closely by the quantum kernel classifiers.

This work introduces a novel application of the Algebraic Dynamic Multilevel (ADM) method for simulating CO₂ storage in deep saline aquifers. By integrating a fully implicit coupling strategy, fully compositional thermodynamics, and adaptive mesh refinement, the ADM framework effectively models phenomena such as buoyancy-driven migration, convective dissolution, and phase partitioning under various subsurface conditions. The method starts with the construction of a hierarchy of multilevel grids and the generation of localized multiscale basis functions, which account for heterogeneities at each coarse level. During the simulation, the ADM method dynamically refines areas with significant overall CO₂ mass fraction gradients while coarsening smooth regions, thus optimizing computational resources without compromising the accuracy required to capture essential flow and transport characteristics. This dynamic grid adjustment is facilitated by algebraic prolongation and restriction operators, which map the fine-scale system onto a coarser grid suited to the evolving distribution of the CO₂ plume. This feature allows the ADM to navigate various coarsening thresholds efficiently, striking a trade-off between computational economy and detailed simulation accuracy. Even at relatively higher thresholds, key trapping mechanisms are captured with sufficient detail for quantification. These capabilities make the ADM framework well suited for long-term CO₂ sequestration in highly heterogeneous reservoirs, where large-scale models may otherwise become impractically expensive, offering a practical balance between the need for detailed simulations and manageable computational requirements. Overall, the ADM framework proves to be a robust tool for designing, monitoring, and analyzing large-scale CO₂ storage operations, supporting reliable and cost-effective solutions in carbon management.

This work extends our previous study from S. Shrestha et al. (2024) by introducing a new abstract framework for Variational Multiscale (VMS) methods at the discrete level. We introduce the concept of what we define as the optimal projector and present a discretisation approach that yields a numerical solution closely approximating the optimal projection of the infinite-dimensional continuous solution. In this approach, the infinite-dimensional unresolved scales are approximated in a finite-dimensional subspace using the numerically computed Fine-Scale Greens’ function of the underlying symmetric problem. The proposed approach involves solving the VMS problem on two separate meshes: a coarse mesh for the full PDE and a fine mesh for the symmetric part of the continuous differential operator. We consider the 1D and 2D steady advection–diffusion problems in both direct and mixed formulations as the test cases in this paper. We first present an error analysis of the proposed approach and show that the projected solution is achieved as the approximate Greens’ function converges to the exact one. Subsequently, we demonstrate the working of this method where we show that it can exponentially converge to the chosen optimal projection. We note that the implementation of the present work employs the Mimetic Spectral Element Method (MSEM), although, it may be applied to other Finite/Spectral Element or Isogeometric frameworks. Furthermore, we propose that VMS should not be viewed as a stabilisation technique; instead, the base scheme should be inherently stable, with VMS enhancing the solution quality by supplementing the base scheme.

CO₂ capture and storage is a viable solution in the effort to mitigate global climate change. Deep saline aquifers, in particular, have emerged as promising storage options, owing to their vast capacity and widespread distribution. However, the task of proficiently monitoring and simulating CO₂ behavior within these formations poses significant challenges. To address this, we introduce the physics-constraint neural network for CO₂ storage (CO₂PCNet), a model specifically designed for simulating and monitoring CO₂ storage in deep saline aquifers during injection and post-injection periods. Recognizing the significant challenges in accurately modeling the distribution and movement of CO₂ under varying permeability conditions, the CO₂PCNet integrates the principles of physics with the robustness of deep learning, serving as a powerful surrogate model. The architecture of CO₂PCNet starts with an encoder that adeptly processes spatial features from overall mole fraction (z_CO2) and pressure fields (P_l), capturing the complex dynamics of a CO₂ trajectory. By incorporating permeability information through a conditioning step, the network ensures a faithful representation of the influences on CO₂ behavior in subsurface conditions. A ConvLSTM module subsequently discerns temporal evolutions, reflecting the real-world progression of CO₂ plumes within the reservoir. Lastly, the decoder precisely reconstructs the predictive spatial profile of CO₂ distribution. CO₂PCNet, with its integration of convolutional layers, recurrent mechanisms, and physics-informed constraints, offers a refined approach to CO₂ storage simulation. This model offers the potential of utilizing advanced computational methods in advancing CCS practices.

In this paper, we build on the work of Hughes and Sangalli (2007) dealing with the explicit computation of the Fine-Scale Greens’ function. The original approach chooses a set of functionals associated with a projector to compute the Fine-Scale Greens’ function. The construction of these functionals, however, does not generalise to arbitrary projections, higher dimensions, or Spectral Element methods. We propose to generalise the construction of the required functionals by using dual functions. These dual functions can be directly derived from the chosen projector and are explicitly computable. We show how to find the dual functions for both the L² and the H₀¹ projections. We then go on to demonstrate that the Fine-Scale Greens’ functions constructed with the dual basis functions consistently reproduce the unresolved scales removed by the projector. The methodology is tested using one-dimensional Poisson and advection–diffusion problems, as well as a two-dimensional Poisson problem. We present the computed components of the Fine-Scale Greens’ function, and the Fine-Scale Greens’ function itself. These results show that the method works for arbitrary projections, in arbitrary dimensions. Moreover, the methodology can be applied to any Finite/Spectral Element or Isogeometric framework.

In this work, we present a mass, energy, enstrophy and vorticity conserving (MEEVC) mixed finite element discretization for two-dimensional incompressible Navier-Stokes equations as an alternative to the original MEEVC scheme proposed in A. Palha and M. Gerritsma (2017) [5]. The present method can incorporate general boundary conditions. Conservation properties are proven. Supportive numerical experiments with both exact and inexact quadrature are provided.

CO₂ sequestration and storage in deep saline aquifers is a promising technology for mitigating the excessive concentration of the greenhouse gas in the atmosphere. However, accurately predicting the migration of CO₂ plumes requires complex multi-physics-based numerical simulation approaches, which are prohibitively expensive due to highly nonlinear coupled governing equations and uncertainties in heterogeneous spatial parameter distributions. To address this challenge, we developed an end-to-end deep learning workflow employing encoder–decoder architectures with residual network (ResNet) to efficiently predicts the spatial–temporal evolution of the solution CO₂-brine ratio (R_s) and gas saturation (S_g) – the two essential tasks for quantifying the amount of trapped CO₂ – given heterogeneous permeability fields as input. Specifically, we introduce a general multi-task learning with consistency (MTLC) framework to simultaneously predict R_s and S_g. The MTLC model leverages related tasks with less computational expensive labeled datasets to improve generalization ability. In our study, predictions for multiple tasks from the same permeability realization are not independent but expected to be consistent, as the proposed framework utilizes data-driven cross-task consistency constraints to augment learning of related tasks. Our deep learning model is trained based on physical trapping mechanisms, which play a dominant role in the CO₂ migration process. The results demonstrate that the MTLC model with joint learning yields more accurate predictions and improved generalization for predicting CO₂ migration in several test cases. Furthermore, our workflow is 10⁵ times faster than a high-fidelity physics-based numerical simulator, making it a viable alternative for field-scale applications.

In this work we use algebraic dual spaces with a domain decomposition method to solve the Darcy equations. We define the broken Sobolev spaces and their finite dimensional counterparts. A global trace space is defined that connects the solution between the broken spaces. Use of algebraic dual spaces results in a sparse, metric-free representation of the incompressibility constraint, the pressure gradient term, and on the continuity constraint between the sub domains. To demonstrate this, we solve two test cases: (i) a manufactured solution case, and (ii) an industrial benchmark reservoir modelling problem SPE10. The results demonstrate that the dual spaces can be used for domain decomposition formulation, and despite having more unknowns, requires less simulation time compared to the continuous Galerkin formulation, without compromising on the accuracy of the solution.

Several investigations have been undertaken to study the velocity and temperature fields associated with the thermal mixing between fluids, and resulting thermal striping in a T-junction. However, the available experimental databases are not sufficient to describe the involved physics in adequate detail, and, due to experimental limitations, accurate data on velocity and temperature fluctuations in regions close to the wall are not available. Computational Fluid Dynamics (CFD) can play an important role in predicting such complex flow features. However, predicting complex thermal fatigue phenomena is a challenge for the available momentum and heat flux turbulence models. Furthermore, such models need to be extensively validated. The aim of the present work is to design a reference numerical experiment for Direct Numerical Simulation (DNS) of a thermal fatigue scenario using Reynolds-Averaged Navier-Stokes (RANS) simulations. First, the feasibility of scaling down the Reynolds number from experimental cases to a computationally-feasible range is investigated. The junction corner shape is also modified to a slightly rounded corner, ensuring that the underlying fundamental physical phenomena of turbulence and thermal mixing flow features are preserved. Finally, the pipe lengths of the model were calibrated to ensure there would be no interference of the upstream developing region and the outlet boundary conditions on the thermal mixing at the junction. A sample under-resolved DNS case, with unity and low-Prandtl number passive temperature scalars, with iso-temperature, iso-flux and mixed (Robin) wall boundary conditions, are presented. This proof-of-concept simulation contributes to the finalization of the set-up for fully-resolved DNS with respect to the computational grid size selection and transient characteristics.

We introduce a mimetic dual-field discretization which conserves mass, kinetic energy and helicity for three-dimensional incompressible Navier-Stokes equations. The discretization makes use of a conservative dual-field mixed weak formulation where two evolution equations of velocity are employed and dual representations of the solution are sought for each variable. A temporal discretization, which staggers the evolution equations and handles the nonlinearity such that the resulting discrete algebraic systems are linear and decoupled, is constructed. The spatial discretization is mimetic in the sense that the finite dimensional function spaces form a discrete de Rham complex. Conservation of mass, kinetic energy and helicity in the absence of dissipative terms is proven at the discrete level. Proper dissipation rates of kinetic energy and helicity in the viscous case are also proven. Numerical tests supporting the method are provided.

We introduce a domain decomposition structure-preserving method based on a hybrid mimetic spectral element method for three-dimensional linear elasticity problems in curvilinear conforming structured meshes. The method is an equilibrium method which satisfies pointwise equilibrium of forces. The domain decomposition is established through hybridization which first allows for an inter-element normal stress discontinuity and then enforces the normal stress continuity using a Lagrange multiplier which turns out to be the displacement in the trace space. Dual basis functions are employed to simplify the discretization and to obtain a higher sparsity. Numerical tests supporting the method are presented.

In ℝⁿ, let Λ^k(Ω) represent the space of smooth differential k-forms in Ω. The de Rham complex consists of a sequence of spaces, Λ^k(Ω), k = 0, 1…, n, connected by the exterior derivative, d: Λ^k(Ω) → Λ^k+1(Ω). Appropriately chosen B-spline spaces together with their associated dual B-spline spaces form a discrete double de Rham complex. In practical applications, this discrete double de Rham complex leads to very sparse systems. In this paper, this construction will be explained and illustrated by means of a non-trivial three-dimensional example.

In this paper we present a discontinuous least-squares spectral element method for Stokes equations with primitive variable formulation on both smooth and non-smooth domains. We propose an exponentially accurate numerical scheme based on the stability estimates and implement it on different polygonal domains. Preconditioned conjugate gradient method is used to obtain the numerical solution.

We present a hybrid mimetic spectral element formulation for Darcy flow. The discrete representations for (1) conservation of mass, and (2) inter-element continuity, are topological relations that lead to sparse matrix systems. These constraints are independent of the element size and shape, and thus invariant under mesh transformations. The resultant algebraic system is extremely sparse even for high degree polynomial basis. Furthermore, the system can be efficiently assembled and solved for each element separately.

Given a sequence of finite element spaces which form a de Rham sequence, we will construct dual representations of these spaces with associated differential operators which connect these spaces such that they also form a de Rham sequence. The dual representations also need to satisfy the de Rham sequence on the domain boundary. The matrix which converts primal representations to dual representations – the Hodge matrix – is the mass or Gram matrix. It will be shown that a bilinear form of a primal and a dual representation is equal to the vector inner product of the expansion coefficients (degrees of freedom) of both representations. This leads to very sparse system matrices, even for high order methods. The differential operators for dual representations will be defined. Vector operations, grad, curl and div, for primal and dual representations are both topological and do not depend on the metric, i.e. the size and shape of the mesh or the order of the numerical method. Derivatives are evaluated by applying sparse incidence and inclusion matrices to the expansion coefficients of the representations. As illustration of the use of dual representations, the method will be applied to (i) a mixed formulation for the Poisson problem in 3D, (ii) it will be shown that this approach allows one to preserve the equivalence between Dirichlet and Neumann problems in the finite dimensional setting and, (iii) the method will be applied to the approximation of grad–div eigenvalue problem on affine and non-affine meshes.

In this paper we will consider two curl-curl equations in two dimensions. One curl-curl problem for a scalar quantity F and one problem for a vector field E. For Dirichlet boundary conditions n× E= Ê_⊣ on E and Neumann boundary conditions n×curlF=Ê⊣, we expect the solutions to satisfy E = curl F. When we use algebraic dual polynomial representations, these identities continue to hold at the discrete level. Equivalence will be proved and illustrated with a computational example.

In this paper, we present a hybrid mimetic method which solves the mixed formulation of the Poisson problem on curvilinear quadrilateral meshes. The method is hybrid in the sense that the domain is decomposed into multiple disjoint elements and the interelement continuity is enforced using a Lagrange multiplier. The method is mimetic in the sense that the discrete divergence operator is exact. By using the mimetic basis functions and their dual representations, various metric-free discrete terms are obtained. The discrete system can be efficiently solved by first solving a reduced system for the Lagrange multiplier. Numerical experiments which validate the method are presented.

In this paper, we will show that the equivalence of a div-grad Neumann problem and a grad-div Dirichlet problem can be preserved at the discrete level in 3-dimensional curvilinear domains if algebraic dual polynomial representations are employed. These representations will be introduced. Proof of the equivalence at the discrete level follows from the construction of the algebraic dual representations. A 3-dimensional test problem in curvilinear coordinates will illustrate this approach.