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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., Aerodynamics, Numerical Analysis, Ship Hydromechanics and Structures

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., Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public., Aerodynamics

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 (Rs) and gas saturation (Sg) – 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 Rs and Sg. 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., Aerodynamics, Reservoir Engineering

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., Aerodynamics

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., Structural Optimization and Mechanics, Aerodynamics

Structural mechanics is commonly modeled by (systems of) partial differential equations (PDEs). Except for very simple cases where analytical solutions exist, the use of numerical methods is required to find approximate solutions. However, for many problems of practical interest, the computational cost of classical numerical solvers running on classical, that is, silicon-based computer hardware, becomes prohibitive. Quantum computing, though still in its infancy, holds the promise of enabling a new generation of algorithms that can execute the most cost-demanding parts of PDE solvers up to exponentially faster than classical methods, at least theoretically. Also, increasing research and availability of quantum computing hardware spurs the hope of scientists and engineers to start using quantum computers for solving PDE problems much faster than classically possible. This work reviews the contributions that deal with the application of quantum algorithms to solve PDEs in structural mechanics. The aim is not only to discuss the theoretical possibility and extent of advantage for a given PDE, boundary conditions and input/output to the solver, but also to examine the hardware requirements of the methods proposed in literature., Aerospace Structures & Computational Mechanics, Numerical Analysis, Aerodynamics

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., Computational Design and Mechanics, Aerodynamics

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., Aerodynamics

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., Aerodynamics

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., Aerodynamics

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., Aerodynamics

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., Aerodynamics

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., Aerodynamics

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., Aerodynamics

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., Aerodynamics

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., Aerodynamics

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., Aerodynamics

In this paper the spectral mimetic least-squares method is applied to a two-dimensional div-curl system. A test problem is solved on orthogonal and curvilinear meshes and both h- and p-convergence results are presented. The resulting solutions will be pointwise divergence-free for these test problems. For N> 1 optimal convergence rates on an orthogonal and a curvilinear mesh are observed. For N= 1 the method does not converge., Aerodynamics

In this paper, an h∕p spectral element method with least-square formulation for parabolic interface problem will be presented. The regularity result of the parabolic interface problem is proven for non-homogeneous interface data. The differentiability estimates and the main stability estimate theorem, using non-conforming spectral element functions, are proven. Error estimates are derived for h and p versions of the proposed method. Specific numerical examples are given to validate the theory., Aerodynamics

In this paper the spectral mimetic least-squares method is applied to a two-dimensional div-curl system. A test problem is solved on orthogonal and curvilinear meshes and both h- and p-convergence results are presented. The resulting solutions will be pointwise divergence-free for these test problems. For N> 1 optimal convergence rates on an orthogonal and a curvilinear mesh are observed. For N= 1 the method does not converge., Aerodynamics

In this paper, an h∕p spectral element method with least-square formulation for parabolic interface problem will be presented. The regularity result of the parabolic interface problem is proven for non-homogeneous interface data. The differentiability estimates and the main stability estimate theorem, using non-conforming spectral element functions, are proven. Error estimates are derived for h and p versions of the proposed method. Specific numerical examples are given to validate the theory., Aerodynamics

We present a spectral mimetic least-squares method which is fully conservative and decouples the primal and dual variables., Aerodynamics, Wind Energy & Propulsion, Aerospace Engineering

We present a spectral mimetic least-squares method which is fully conservative and decouples the primal and dual variables., Aerodynamics, Wind Energy & Propulsion, Aerospace Engineering

Aerodynamics, Wind Energy & Propulsion, Aerospace Engineering

Aerodynamics, Wind Energy & Propulsion, Aerospace Engineering

Aerospace Engineering

Aerospace Engineering

Aerospace Engineering

Aerospace Engineering

Least-squares methods for partial differential equations are based on a norm-equivalence between the error norm and the residual norm. The resulting algebraic system of equations, which is symmetric positive definite, can also be obtained by solving a weighted collocation scheme using least-squares to solve the resulting algebraic equations. Furthermore, least-squares allows to ad extra constraints to the system. In the present work the entropy is added as an extra inequality constraint to ensure only physical solutions for the one-dimensional inviscid Burgers equation are obtained.

For goal-oriented model adaptation a model-error estimator is required to drive the adaptation process. In recent years publications have appeared on the dual-weighted residual (DWR) method in the application of model-error estimation in output functionals. In this paper we study the application of the DWR method for convection-diffusion problems where hierarchical models are of different type. Omitting the diffusion operator often results in a singular perturbation problem considering the model residual in the limit of vanishing diffusion. This is caused by the change of mathematical type of the model equations and therefore the applied boundary conditions. In this work we show how a model error estimator is developed for steady and unsteady convection-diffusion problems. It is found that a weak formulation and weakly imposing boundary and initial conditions leads to a dual-weighted model-error estimator that also incorporates boundary residuals.

In this work a new approach to time dependent problems in combination with the Least-Squares Spectral Element Method (LSQSEM) will be discussed. Various time-stepping formulations will be presented. These time-stepping formulations will be compared to the full space-time formulation. It will be shown that time-stepping formulations give accurate results for comparable CPU times. Furthermore is will be shown that a smaller timestep or a higher polynomial degree not always decreases the error norm.

This paper describes the use of the Least-Squares Spectral Element Method for non-linear hyperbolic equations. The one-dimensional inviscid Burgers equation is specifically subject of investigation. A second order backward difference method is used for time stepping. The behaviour of this formulation is examined by application to a testcase where a moving shock develops. For this testcase an hp-convergence study is performed.

This papers describes the use of the Least-Squares Spectral Element Method to polynomial Chaos to solve stochastic partial differential equations. The method will be described in detail and a comparison will be presented between the least-squares projection and the conventional Galerkin projection.

This paper discusses the use of the Least-Squares Spectral Element Method in solving the linear, 1-dimensional advection-reaction equation. Well-posedness of the Least-Squares formulation will be derived. The formulation and its results will be compared to the standard Galerkin Spectral Element Method.

This papers describes the use of the Least-Squares Spectral Element Method to polynomial Chaos to solve stochastic partial differential equations. The method will be described in detail and a comparison will be presented between the least-squares projection and the conventional Galerkin projection., Aerospace Engineering

Least-squares methods for partial differential equations are based on a norm-equivalence between the error norm and the residual norm. The resulting algebraic system of equations, which is symmetric positive definite, can also be obtained by solving a weighted collocation scheme using least-squares to solve the resulting algebraic equations. Furthermore, least-squares allows to ad extra constraints to the system. In the present work the entropy is added as an extra inequality constraint to ensure only physical solutions for the one-dimensional inviscid Burgers equation are obtained., Aerospace Engineering

In this work a new approach to time dependent problems in combination with the Least-Squares Spectral Element Method (LSQSEM) will be discussed. Various timestepping formulations will be presented. These time-stepping formulations will be compared to the full space-time formulation. It will be shown that time-stepping formulations give accurate results for comparable CPU times. Furthermore is will be shown that a smaller timestep or a higher polynomial degree not always decreases the error norm., Aerospace Engineering

This paper discusses the use of the Least-Squares Spectral Element Method in solving the linear, 1-dimensional advection-reaction equation. Well-posedness of the Least-Squares formulation will be derived. The formulation and its results will be compared to the standard Galerkin Spectral Element Method., Aerospace Engineering

For goal-oriented model adaptation a model-error estimator is required to drive the adaptation process. In recent years publications have appeared on the dual-weighted residual (DWR) method in the application of model-error estimation in output functionals. In this paper we study the application of the DWR method for convection-diffusion problems where hierarchical models are of different type. Omitting the diffusion operator often results in a singular perturbation problem considering the model residual in the limit of vanishing diffusion. This is caused by the change of mathematical type of the model equations and therefore the applied boundary conditions. In this work we show how a model error estimator is developed for steady and unsteady convection-diffusion problems. It is found that a weak formulation and weakly imposing boundary and initial conditions leads to a dual-weighted model-error estimator that also incorporates boundary residuals., Aerospace Engineering

For goal-oriented model adaptation a model-error estimator is required to drive the adaptation process. In recent years publications have appeared on the dual-weighted residual (DWR) method in the application of model-error estimation in output functionals. In this paper we study the application of the DWR method for convection-diffusion problems where hierarchical models are of different type. Omitting the diffusion operator often results in a singular perturbation problem considering the model residual in the limit of vanishing diffusion. This is caused by the change of mathematical type of the model equations and therefore the applied boundary conditions. In this work we show how a model error estimator is developed for steady and unsteady convection-diffusion problems. It is found that a weak formulation and weakly imposing boundary and initial conditions leads to a dual-weighted model-error estimator that also incorporates boundary residuals.

In this work a new approach to time dependent problems in combination with the Least-Squares Spectral Element Method (LSQSEM) will be discussed. Various time-stepping formulations will be presented. These time-stepping formulations will be compared to the full space-time formulation. It will be shown that time-stepping formulations give accurate results for comparable CPU times. Furthermore is will be shown that a smaller timestep or a higher polynomial degree not always decreases the error norm.

This paper describes the use of the Least-Squares Spectral Element Method for non-linear hyperbolic equations. The one-dimensional inviscid Burgers equation is specifically subject of investigation. A second order backward difference method is used for time stepping. The behaviour of this formulation is examined by application to a testcase where a moving shock develops. For this testcase an hp-convergence study is performed.

Least-squares methods for partial differential equations are based on a norm-equivalence between the error norm and the residual norm. The resulting algebraic system of equations, which is symmetric positive definite, can also be obtained by solving a weighted collocation scheme using least-squares to solve the resulting algebraic equations. Furthermore, least-squares allows to ad extra constraints to the system. In the present work the entropy is added as an extra inequality constraint to ensure only physical solutions for the one-dimensional inviscid Burgers equation are obtained.

This papers describes the use of the Least-Squares Spectral Element Method to polynomial Chaos to solve stochastic partial differential equations. The method will be described in detail and a comparison will be presented between the least-squares projection and the conventional Galerkin projection., Aerospace Engineering

Least-squares methods for partial differential equations are based on a norm-equivalence between the error norm and the residual norm. The resulting algebraic system of equations, which is symmetric positive definite, can also be obtained by solving a weighted collocation scheme using least-squares to solve the resulting algebraic equations. Furthermore, least-squares allows to ad extra constraints to the system. In the present work the entropy is added as an extra inequality constraint to ensure only physical solutions for the one-dimensional inviscid Burgers equation are obtained., Aerospace Engineering

This papers describes the use of the Least-Squares Spectral Element Method to polynomial Chaos to solve stochastic partial differential equations. The method will be described in detail and a comparison will be presented between the least-squares projection and the conventional Galerkin projection.

This paper discusses the use of the Least-Squares Spectral Element Method in solving the linear, 1-dimensional advection-reaction equation. Well-posedness of the Least-Squares formulation will be derived. The formulation and its results will be compared to the standard Galerkin Spectral Element Method.

In this work a new approach to time dependent problems in combination with the Least-Squares Spectral Element Method (LSQSEM) will be discussed. Various timestepping formulations will be presented. These time-stepping formulations will be compared to the full space-time formulation. It will be shown that time-stepping formulations give accurate results for comparable CPU times. Furthermore is will be shown that a smaller timestep or a higher polynomial degree not always decreases the error norm., Aerospace Engineering

This paper describes the use of the Least-Squares Spectral Element Method for non-linear hyperbolic equations. The one-dimensional inviscid Burgers equation is specifically subject of investigation. A second order backward difference method is used for time stepping. The behaviour of this formulation is examined by application to a testcase where a moving shock develops. For this testcase an hp-convergence study is performed., Aerospace Engineering