Adaptive Mesh Refinement Strategy Research Articles

Optimization of Convergence of Mixed Finite Element Approximations Analysis

The numerical approximation of partial differential equations (PDEs) plays a vital role in many scientific and engineering applications. Mixed finite element methods have emerged as powerful techniques for solving a wide range of PDEs due to their ability to handle problems with mixed variables, such as fluid flow and elasticity. However, ensuring the convergence of mixed finite element approximations remains a challenging task. It presents a comprehensive analysis of the optimization strategies employed to enhance the convergence of mixed finite element approximations. We investigate the key factors that impact the convergence behaviour of these methods and propose various techniques to mitigate convergence issues. we discuss the importance of appropriate discretization strategies for mixed finite element approximations. We analyze the impact of element types, mesh refinement, and stabilization techniques on the convergence rates. By examining the properties of the underlying mixed variational formulations, we identify the optimal discretization choices that lead to improved convergence behaviour. We delve into the analysis of numerical stability and consistency in the context of mixed finite element methods. We explore the role of stabilization techniques, such as bubble functions and penalty terms, in mitigating instabilities and achieving optimal convergence rates. We investigate the effect of different stabilization parameters and establish guidelines for their selection to ensure both stability and convergence. We address the issue of error estimation and adaptivity in mixed finite element approximations. We review error indicators and adaptive mesh refinement strategies that enable the refinement of regions with high solution gradients, thus enhancing the convergence rates. We discuss the interplay between error estimation and adaptive refinement and present numerical examples illustrating their effectiveness.
 We highlight recent advancements in optimization algorithms specifically tailored for enhancing convergence in mixed finite element approximations. We explore strategies like multigrid methods, preconditioning techniques, and domain decomposition methods, which accelerate the convergence rates and enable the solution of large-scale problems. This paper provides a comprehensive analysis of the optimization of convergence for mixed finite element approximations. It serves as a valuable resource for researchers and practitioners seeking to improve the efficiency and accuracy of numerical solutions obtained through mixed finite element methods.

A consistent multi-level subgrid scale closure for large eddy simulation of compressible flow using adaptive mesh refinement

A novel subgrid closure model is proposed for performing Large Eddy Simulation (LES) of turbulent compressible flows with block structured Adaptive Mesh Refinement (AMR) strategy. The new approach, henceforth called as AMRLES, is an improvement over the well-established one-equation based subgrid turbulent kinetic energy model with corrections proposed to consistently handle the total resolved turbulent kinetic energy across a multi-level AMR grid hierarchy. The proposed correction for the subgrid kinetic energy is based on an explicit on-the-fly filtering operation from finer grids to underlying coarser grids. The effect of the inter-grid correction is demonstrated by investigating advection of decaying isotropic turbulence across a coarse/fine and fine/coarse AMR grid interface. In the case of fine-to-coarse transition, the multi-level model is shown to correctly recover the subgrid turbulent kinetic energy. AMRLES is then applied to study the canonical problem of shock turbulent interaction. Two different Reynolds numbers are considered and the behavior of the inter-grid corrections on the subgrid turbulent kinetic energy and the subgrid turbulent stresses are analyzed for both the cases. Comparison with filtered DNS results indicates that AMRLES is able to predict the behavior of turbulent statistics across the shock in a satisfactory manner.

An algorithm for adaptive introduction and arrangement of velocity discontinuities within 3D finite-element-based upper bound limit analysis approaches

This paper presents a new adaptive strategy to efficiently exploit velocity discontinuities in 3D finite-element-based upper bound limit analysis formulations. Based on an initial upper bound result, obtained with a conventional approach without velocity discontinuities, possible planes of plastic flow localisation are determined at each strain-rate evaluation node and, subsequently, this information is used to sequentially introduce discontinuities into the considered discretised structure. During a few iterations, by means of introducing new and adjusting existing discontinuities, an optimal velocity discontinuity layout is obtained. For the general 3D case, the geometry of this layout is defined by the well-known level set method, standardly used to define the geometry of cracks in the extended finite element method.To make this method also applicable for orthotropic strength behaviours, traction-based yield functions defining the plastic flow across discontinuities are derived from their stress-based counterparts. This procedure is outlined in detail and the obtained traction-based yield functions are verified numerically, to guarantee a consistent strength behaviour throughout the whole discretised structure.By means of three different examples, including isotropic as well as orthotropic yield functions, the performance of the proposed strategy is investigated and upper bound results as well as failure modes are compared to reference solutions. The proposed approach delivers reliable upper bounds for each example and the majority of plastic flow takes place across the sensibly-arranged discontinuities. For this reason, very good upper bounds can be obtained with a quite coarse finite element mesh and only few introduced velocity discontinuities. This represents an attractive alternative to commonly-used adaptive mesh refinement strategies, where often a huge number of degrees of freedom need to be added to capture localised failure.

Retarded potentials for wave propagation in elastic trusses

AbstractA space‐time boundary element method for the dynamic simulation of simplified elastic truss systems is proposed. The considered truss systems consist of elastic rods whose dynamic behaviour is governed solely by the 1D wave equation. By using time domain boundary integral equations the problem reduces to the nodes of the truss system and therefore no spatial discretization is necessary. The discretization employs an adaptive mesh refinement scheme enabling the accurate resolution of non‐smooth solutions.

Ultrahigh-resolution model of a breakout CME embedded in the solar wind

Aims. We investigate the effect of a background solar wind on breakout coronal mass ejections, in particular, the effect on the different current sheets and the flux rope formation process. Methods. We obtained numerical simulation results by solving the magnetohydrodynamics equations on a 2.5D (axisymmetric) stretched grid. Ultrahigh spatial resolution is obtained by applying a solution adaptive mesh refinement scheme by increasing the grid resolution in regions of high electrical current, that is, by focussing on the maximum resolution of the current sheets that are forming. All simulations were performed using the same initial base grid and numerical schemes; we only varied the refinement level. Results. A background wind that causes a surrounding helmet streamer has been proven to have a substantial effect on the current sheets that are forming and thus on the dynamics and topology of the breakout release process. Two distinct ejections occur: first, the top of the helmet streamer detaches, and then the central arcade is pinched off behind the top of the helmet streamer. This is different from the breakout scenario that does not take the solar wind into account, where only the central arcade is involved in the eruption. In the new ultrahigh-resolution simulations, small-scale structures are formed in the lateral current sheets, which later merge with the helmet streamer or reconnect with the solar surface. We find that magnetic reconnections that occur at the lateral breakout current sheets deliver the major kinetic energy contribution to the eruption and not the reconnection at the so-called flare current sheet, as was seen in the case without background solar wind.

A-SLEIPNNIR: A multiscale, anisotropic adaptive, particle level set framework for moving interfaces. Transport equation applications

We introduce in this paper ‘A-SLEIPNNIR’ (“Adaptive Semi-Lagrangian Ensemble Implementation of Particle level set for Newtonian and non-Newtonian Interfacial Rheology”). The proposed method combines a semi-Lagrangian approach for the transport operator, a particle level set technique for interface capturing featuring second-order accurate redistancing, and anisotropic mesh refinement for spatial resolution. The high-order, quadratic Finite Element discretization, together with an a posteriori error analysis produce a metric tensor that is passed on to an anisotropic mesh generator. We explore the peculiar features of the numerical scheme such as the impact of the marker particles, the reinitialization procedure or the Adaptive Mesh Refinement (AMR) strategy, in a series of benchmark problems for the solution of the interface transport equation with analytically supplied, divergence-free velocity fields. The results show that our approach yields an accurate and robust, yet computationally inexpensive framework for moving interfaces suitable for Scientific and Engineering applications.

A numerical study of liquid atomization regimes by means of conservative level-set simulations

In this work, a conservative level-set finite-volume solver for interface-capturing is employed to perform the direct numerical simulation of Liquid Jets discharging into a quiescent air chamber. The scope is to propose a complete break-up regimes map entirely obtained from numerical simulations. The solver accounts for an adaptive mesh refinement strategy to optimize the computational resources. The numerical model is firstly validated in the context of 3D atomization by simulating the behavior of 3D Coaxial Liquid-Air Jets. Next, we propose an overview of the physical behavior of Liquid Jets, mainly proceeding from theoretical and experimental works. Hence, we perform a series of simulations aimed at studying the variability of the Liquid Jet characteristics as function of selected input parameters. In particular, the analyzed cases are characterized by variable values of Reynolds, Ohnesorge and Weber numbers. The patterns obtained in simulations are compared to the ones expected from bibliographic studies, situating each case on a break-up regime map. A general good agreement is found in the identification of the various break-up regimes and characteristic lengths, while the major differences have been highlighted and interpreted.

A posteriori error estimation and adaptivity based on VMS for the Stokes problem

SummaryIn this paper, we present residual‐based error estimators applied to the Stokes problem. Implicit and explicit error estimators are developed to predict the error of the numerical solution obtained by a stabilized finite element formulation. Both error estimators arise from the variational multiscale framework, in which the variational form is split into coarse scales (finite element method solution) and fine scales (committed error). This leads to a local problem set on each element in which the error and the residuals are involved. The different ways of linking the error to the residuals produce two error estimators. Local and global error estimates measured in the H1‐seminorm are provided for triangles and quadrilaterals. As an application, using the local error estimates, an adaptive mesh refinement strategy is implemented in order to optimize the computational resources.

Surface wettability effect on the rising of a bubble attached to a vertical wall

This paper deals with the dynamics of rising bubbles attached to a vertical wall under different wettability conditions. Even though, bubbles rising freely in a liquid have extensively been studied, bubbles attached to a wall have not been fully understood. Therefore, in this work, rising bubbles attached to a vertical wall were numerically investigated by applying the ALE method along with adaptive mesh refinement schemes to properly resolve the bubble interface and its deformation. To consider wall wettability effects, different contact angles of 45°, 90°, and 105° were considered along with the case of freely rising bubbles. The problem was carried out at different Bond numbers of 0.27, 0.4, and 0.635 for different Reynolds numbers 250, 1000, and 2000. It was shown that in contrast to freely rising bubbles, which have a smooth monotone deformation and motion behavior during the rising, attached bubbles experiences different acceleration/deceleration fields with complicated oscillations and deformations. The results show that wall hydrophobicity strongly affects the bubble profile and consequently the dynamics of bubble rising. In addition, depending on the wall wettability and Bond numbers, the wall-bounded bubbles are influenced by different forces leading to different profiles and hence rise up with different velocities along the wall. The results illustrate that bubbles with contact angles less than 90° move with a speed even higher than freely rising bubbles.

Optimal electric power generation with underwater kite systems

In this article we investigate the problem of generating electricity through an underwater kite power system (UKPS). For this problem, we develop the dynamical model for the UKPS and we formulate an optimal control problem to devise the trajectories and controls of the kite that maximize the total energy produced in a given time interval. This is a highly nonlinear problem for which the optimization is challenging. We also develop a numerical solution scheme for the optimal control problem based on direct methods and on adaptive time-mesh refinement. We report results that show that the problem can be quickly solved with a high level of accuracy when using our adaptive mesh refinement strategy. The results provide a set of output power values for different design choices and confirm that electrical energy that can be produced with such device.

Dynamic adaptive mesh refinement of fluid structure interaction using immersed boundary method with two stage corrections

The application of the immersed boundary method (IBM) coupled with adaptive mesh refinement (AMR) is considered one of the powerful tools for solving complex viscous incompressible flow problems. In this paper, the IBM was combined with AMR to solve complex incompressible and viscous fluid with elastic and rigid body problems concerning large structural deformations. In the IBM, the solid and fluid motions at the interface are united by a body force which can be compared to a fraction of a solid volume. The work aims to develop an automatic adaptive mesh refinement strategy to improve the solution near the fluid-structure interface. This is necessary as the flow field might be significantly affected by the structure; therefore, it was essential to precisely capture the boundary layers. The computational results highlighted the capability of this method to improve the flow resolution near the fluid structure. The proposed approach is validated using 2D numerical examples. The approach is validated in terms of its superior performance. The combined IBM-Adaptive mesh refinement approach showed a promising outcome for the investigated mechanical problem. The performance of the method in achieving a solution within a reasonably low computation time is also commendable.

A Virtual Element Method for 2D linear elastic fracture analysis

This paper presents the Virtual Element Method (VEM) for the modeling of crack propagation in 2D within the context of linear elastic fracture mechanics (LEFM). By exploiting the advantage of mesh flexibility in the VEM, we establish an adaptive mesh refinement strategy based on the superconvergent patch recovery for triangular, quadrilateral as well as for arbitrary polygonal meshes. For the local stiffness matrix in VEM, we adopt a stabilization term which is stable for both isotropic scaling and ratio. Stress intensity factors (SIFs) of a polygonal mesh are discussed and solved by using the interaction domain integral. The present VEM formulations are finally tested and validated by studying its convergence rate for both continuous and discontinuous problems, and are compared with the optimal convergence rate in the conventional Finite Element Method (FEM). Furthermore, the adaptive mesh refinement strategies used to effectively predict the crack growth with the existence of hanging nodes in nonconforming elements are examined.

Stochastic collocation approach with adaptive mesh refinement for parametric uncertainty analysis

The presence of a high-dimensional stochastic input domain with discontinuities poses major computational challenges in analyzing and quantifying the effects of the uncertainties in a physical system. In this paper, we propose a stochastic collocation method with adaptive mesh refinement (SCAMR) to deal with high dimensional stochastic systems with discontinuities. Specifically, the proposed approach uses generalized polynomial chaos (gPC) expansion with Legendre polynomial basis and solves for the gPC coefficients using the least squares method. It also implements an adaptive mesh (element) refinement strategy which checks for abrupt variations in the output based on a low-order gPC approximation error to track discontinuities or non-smoothness. In addition, the proposed method involves a criterion for checking possible dimensionality reduction and consequently, the decomposition of the original high-dimensional problem to a number of lower-dimensional subproblems. Specifically, this criterion checks all the existing interactions between input parameters of a specific problem based on the high-dimensional model representation (HDMR) method, and therefore automatically provides the subproblems which only involve interacting input parameters. The efficiency of the approach is demonstrated using examples of both smooth and non-smooth problems with number of input parameters up to 500, and the approach is compared against other existing algorithms.

An adaptive mesh refinement strategy for finite element solution of the elliptic problem

In this paper we develop an adaptive finite element method for elliptic problems. First, we assume that in each subdomain the norm of the approximation error at the current mesh configuration is bounded by the norm of the approximation error obtained at the previous mesh configuration, for some norm Hs. Then an a-posteriori error estimator is designed based on the approximate solution correction between the solution on the last two mesh configurations. Based on this new error estimator, the element-wise refinement strategy in each subdomain is provided for a given tolerance.A discussion on the choice of the coefficients in the assumption is given for different norm spaces and for different degrees of finite element family. Four 2D numerical benchmark examples of different domains and two 3D numerical benchmark examples are tested to demonstrate the robustness of our method. When possible, our numerical results are also compared to corresponding results from existing methods. All the results show that the proposed method is robust and efficient in terms of the number of degrees of freedom.

An Adaptive Staggered Discontinuous Galerkin Method for the Steady State Convection–Diffusion Equation

Staggered grid techniques have been applied successfully to many problems. A distinctive advantage is that physical laws arising from the corresponding partial differential equations are automatically preserved. Recently, a staggered discontinuous Galerkin (SDG) method was developed for the convection–diffusion equation. In this paper, we are interested in solving the steady state convection–diffusion equation with a small diffusion coefficient $$\epsilon $$ . It is known that the exact solution may have large gradient in some regions and thus a very fine mesh is needed. For convection dominated problems, that is, when $$\epsilon $$ is small, exact solutions may contain sharp layers. In these cases, adaptive mesh refinement is crucial in order to reduce the computational cost. In this paper, a new SDG method is proposed and the proof of its stability is provided. In order to construct an adaptive mesh refinement strategy for this new SDG method, we derive an a-posteriori error estimator and prove its efficiency and reliability under a boundedness assumption on $$h/\epsilon $$ , where h is the mesh size. Moreover, we will present some numerical results with singularities and sharp layers to show the good performance of the proposed error estimator as well as the adaptive mesh refinement strategy.

Trajectory optimization for lunar rover performing vertical takeoff vertical landing maneuvers in the presence of terrain

This study presents a trajectory optimization framework for lunar rover performing vertical takeoff vertical landing (VTVL) maneuvers in the presence of terrain using variable-thrust propulsion. First, a VTVL trajectory optimization problem with three-dimensional kinematics and dynamics model, boundary conditions, and path constraints is formulated. Then, a finite-element approach transcribes the formulated trajectory optimization problem into a nonlinear programming (NLP) problem solved by a highly efficient NLP solver. A homotopy-based backtracking strategy is applied to enhance the convergence in solving the formulated VTVL trajectory optimization problem. The optimal thrust solution typically has a “bang-bang” profile considering that bounds are imposed on the magnitude of engine thrust. An adaptive mesh refinement strategy based on a constant Hamiltonian profile is designed to address the difficulty in locating the breakpoints in the thrust profile. Four scenarios are simulated. Simulation results indicate that the proposed trajectory optimization framework has sufficient adaptability to handle VTVL missions efficiently.

Adaptive mesh refinement for elastic modulus reconstruction in elastography.

Meshes play a crucial role in determining the accuracy of the elastic modulus reconstruction in the elastography when the finite element method is employed. In this article, we propose an adaptive mesh refinement strategy which can ensure the coincidence of the meshes with the shape of the inclusions in the observed tissue. This strategy is based on the intensity distribution of the strain image where the variance of the strain distribution in each element of the meshes is used to measure the homogeneity of the element, that is, the larger the strain variance is the more inhomogeneous the element will be and hence more detailed information will be included in this element. For more accurate reconstruction of such detailed information, mesh refinement procedure is implemented in such elements. Besides, two refinement steps are employed for the reconstruction to improve the fitness of the reconstructed image and the observed tissue. Simulation results show that the two-stage adaptive mesh refinement algorithm performs well without needing any prior information about the internal geometric shape in tissue. Not only Young's moduli of models but also shapes of the inclusions can be reconstructed perfectly and quickly with our proposed method.

An adaptive mesh refinement approach for solving non-Hertzian elastic contact problems

Semi-analytical methods are a common way of solving non-hertzian contact problems when designing mechanical components. These methods require of the discretization of the domain into a set of pressure elements and their accuracy and computational cost are related to the number of elements in which the domain is discretized. But, while the accuracy increases as the pressure element mesh is refined, the computational cost increases quadratically with the number of pressure elements. So in the great majority of the cases, a commitment between accuracy and computational cost must be achieved. In this work, a new approach has been developed to improve the performance of semi-analytical methods for solving contact problems. This approach uses an adaptive mesh refinement strategy, based on the quadtree decomposition of the domain. As a result, the computational cost decreases, while the accuracy of the method remains constant.

Convergence study of the h-adaptive PUM and the hp-adaptive FEM applied to eigenvalue problems in quantum mechanics

In this paper the h-adaptive partition-of-unity method and the h- and hp-adaptive finite element method are applied to eigenvalue problems arising in quantum mechanics, namely, the Schrödinger equation with Coulomb and harmonic potentials, and the all-electron Kohn–Sham density functional theory. The partition-of-unity method is equipped with an a posteriori error estimator, thus enabling implementation of error-controlled adaptive mesh refinement strategies. To that end, local interpolation error estimates are derived for the partition-of-unity method enriched with a class of exponential functions. The efficiency of the h-adaptive partition-of-unity method is compared to the h- and hp-adaptive finite element method. The latter is implemented by adopting the analyticity estimate from Legendre coefficients. An extension of this approach to multiple solution vectors is proposed. Numerical results confirm the theoretically predicted convergence rates and remarkable accuracy of the h-adaptive partition-of-unity approach. Implementational details of the partition-of-unity method related to enforcing continuity with hanging nodes are discussed.

A posteriori error estimators for the Frank-Oseen model of liquid crystals

This paper derives a posteriori error estimators for the nonlinear first-order optimality conditions associated with the Frank-Oseen elastic free-energy model of nematic and cholesteric liquid crystals, where the required unit-length constraint is imposed via either a Lagrange multiplier or penalty method. Furthermore, theory establishing the reliability of the proposed error estimator for the penalty method is presented, yielding a concrete upper bound on the approximation error of discrete solutions. The error estimators herein are composed of readily computable quantities on each element of a finite-element mesh, allowing the formulation of an efficient adaptive mesh refinement strategy. Four elastic equilibrium problems are considered to examine the performance of the error estimators and corresponding adaptive mesh refinements against that of a simple uniform refinement scheme. The adapted grids successfully provide significant reductions in computational work while producing solutions that are highly competitive with those of uniform mesh in terms of constraint conformance and computed free energies.