Mesh Refinement Strategy Research Articles

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.

Importance of Dilatancy on the Evolution of Failure Mechanism of a Strip Footing Resting on Horizontal Ground

This paper reports about a finite element based numerical analysis of the failure mechanism developed beneath a strip footing resting on horizontal ground. The observed failure mechanism is manifested in terms of incremental deviatoric strain patterns. In order to address the effect of dilatancy, both associated and non-associated flow rule had been considered. A series of simulation on foundation models were carried out to examine the influence of mesh refinement schemes and soil shear strength properties on the failure mechanism. The numerical outcomes have been verified against experimental investigations available in literature. It has been observed that angle of dilatancy ( $$\psi$$ ) plays a significant role in influencing the evolution of the failure mechanism. Consideration of non-associated flow rule ( $$\psi$$ ≠ φ, φ is the angle of internal friction) leads to the development of asymmetric failure mechanisms, marked by one-directional soil movement beneath the footing, exhibiting mode-switching phenomenon at- or near-failure conditions. Considering associated flow rule ( $$\psi$$ = φ) results in the evolution of a symmetric failure mechanism at the failure condition of the footing. The present study reveals that beyond $$\psi$$ = 3φ/4, a transition from the asymmetric to symmetric failure mechanism occurs. The study shows that, based on the appreciable agreements of the resulting pressure-settlement curves and the ratio of rightward-to-leftward displacement of the footing, FE simulations conducted with different angles of dilatancy can be effectively used to identify the angle of dilatancy prevalent in an experimental investigation.

A simple adaptive mesh refinement scheme for topology optimization using polygonal meshes

The objective of topology optimization is to find the most efficient distribution of material (optimal topology) in a given domain, subjected to design constraints defined by the user. The quality of the new boundary representation depends on the level of mesh refinement: The greater the number of elements in the mesh, the better is the representation of the optimized structure. However, the use of refined meshes implies a high computational cost, particularly with regard to the numerical solution of the linear systems of equations that arise from the finite element method. This paper proposes a new computational strategy for adaptive local mesh refinement using polygonal finite elements in arbitrary two-dimensional domains. The idea is to perform a mesh refinement in regions of material concentration, mostly in inner and outer boundaries, and a mesh derefinement in regions of low material concentration, such as in internal holes. Thus, it is possible to obtain high-resolution optimal topologies with a relatively low computational cost. Representative examples are presented to demonstrate the robustness and efficiency of the proposed methodology by comparing the results obtained here with results from the literature where refined meshes are kept constant throughout the topology optimization process.

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.

Assessment of failure mechanism of a strip footing on horizontal ground considering flow rules

Numerical investigation of the failure mechanism developed beneath a strip footing resting on non-dilatant cohesionless soil has been investigated in a finite element framework. The observed failure mechanism is manifested in terms of incremental displacement and incremental deviatoric strain patterns. A series of simulation on foundation models were carried out to examine the influence of various factors on the failure mechanism, namely meshing pattern, mesh refinement schemes, soil shear strength properties and embedment of the footing. The numerical outcomes have been verified against sophisticated laboratory investigations available in the literature. The developed failure mechanism is found to be asymmetric and markedly one-directional, the nature being prevalently independent to the variation in the influencing factors. Failure mechanism is found to progressively evolve following a mode-switching phenomenon, exhibiting transition from elastic to plastic deformations at- or near-failure. The deviations of the numerically obtained evolving failure mechanism from the same as pre-assumed in the conventional bearing capacity theories are explored and reported herein.

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.

Adaptive multi-patch isogeometric analysis based on locally refined B-splines

We present an effective adaptive analysis procedure in terms of isogeometric analysis (IGA) based on locally refined (LR) B-splines for two-dimensional elasticity problems. One major advantage of the adopted LR B-splines over B-splines and non-uniform rational B-splines (NURBS) is the local refinement, which makes it highly suitable for adaptive analysis. The Zienkiewicz–Zhu estimation is used on error estimator based local refinement. The local refinement is implemented by the structured mesh refinement strategy according to the posteriori error estimator. The proposed methodology is further extended to deal with multiple patches, which are applied to represent complicated shapes of structures, and the Nitsche’s method is employed for coupling patches among multiple patches in the present adaptive IGA. The algorithm for adaptive IGA on multiple patches is also derived. We provide a series of numerical examples for benchmarks and complicated configurations to demonstrate the flexibility and accuracy of the model as well as the significant enhancement in adaptive IGA based on LR B-splines. A careful comparison of the numerical results suggests that the adaptive local refinement strategy offers faster convergence rate than that of uniform refinement strategy.

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.

The best strategy for local mesh refinement with the PCD method

The best strategy for local mesh refinement with the PCD method

The best strategy for local mesh refinement with the PCD method

We propose in this contribution numerical tests for local mesh refinement with the PCD method. We investigate two rates of local mesh refinement: the ratio 2 and the ratio 3. First we introduce a local mesh refinement by subdividing the elements of the zone to be refined by the ratio 2 in each direction, i. e. each coarse element is subdivided by 4 fine elements. Second, we subdivide the elements of the zone to be refined by the ratio 3 in each direction, i. e. each coarse element is subdivided by 9 fine elements. Here we investigate some numerical experiments to show that the optimal local refinement rate is 2. This is in agreement with the regularity assumptions required for the proposed discretisation.

Generation of surface plasmon-polaritons by edge effects

By using numerical and analytical methods, we describe the generation of fine-scale lateral electromagnetic waves, called surface plasmon-polaritons (SPPs), on atomically thick, metamaterial conducting sheets in two spatial dimensions (2D). Our computations capture the two-scale character of the total field and reveal how each edge of the sheet acts as a source of an SPP that may dominate the diffracted field. We use the finite element method to numerically implement a variational formulation for a weak discontinuity of the tangential magnetic field across a hypersurface. An adaptive, local mesh refinement strategy based on a posteriori error estimators is applied to resolve the pronounced two-scale character of wave propagation and radiation over the metamaterial sheet. We demonstrate by numerical examples how a singular geometry, e.g., sheets with sharp edges, and sharp spatial changes in the associated surface conductivity may significantly influence surface plasmons in nanophotonics.

Computational Design of Acoustic Materials Using an Adaptive Optimization Algorithm

We consider the problem of design of the acoustic structure of arbitrary geometry with prescribed desired properties. We use optimization approach for the solution of this problem and minimize the Tikhonov functional on adaptively refined meshes. These meshes are refined locally only in places where the acoustic structure should be designed. Our special symmetric mesh refinement strategy together with interpolation procedure allows the construction of the symmetric acoustic material with prescribed properties. Efficiency of the presented adaptive optimization algorithm is illustrated on the construction of the symmetric acoustic material in two dimensions.

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.

RBF-FD schemes for option valuation under models with price-dependent and stochastic volatility

Radial basis functions based finite difference schemes for the solution of partial differential equations have the advantage that an optimal choice of the shape parameter can yield better accuracies than standard finite difference discretisations based on the same number of nodal points. Such schemes known as local radial basis functions methods are considered for the pricing of options under the constant elasticity of variance and the Heston stochastic volatility model. A general methodology for approximating first and second order derivative terms in the finance pdes is presented and the resulting schemes are applied for option valuation. For one-dimensional problems, we derive a compact-RBF scheme which achieves a higher order accuracy when combined with a local mesh refinement strategy. Numerical results and comparisons made for European, American and barrier options illustrate the good performances of the localized radial basis functions methods.