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Motivated by the moving mesh method in conventional finite element analysis, in this paper, we propose an r -adaptive isogeometric analysis framework by Winslow’s mapping and monitor function approach. We construct a new monitor function based on the absolute curvature metric of the isogeometric solution surface. In the proposed r -adaptive framework, the redistribution of the inner control points is performed to increase the density of control points in the regions where curvature based or error based monitor functions are maximum. The new locations of the inner control points are obtained by solving governing PDEs using isogeometric collocation method which is more efficient in solving large-scale problems. By employing Winslow’s mapping we ensure that the generated r -adaptive parameterization is valid for isogeometric analysis and can be used as optimal initial grid for subsequent h -refinements. Multiple two-dimensional heat conduction examples are solved which confirm the effectiveness of the proposed method.

In this paper a hybrid difference method is used to approximate a parameterized singular perturbation problem. An improved a posteriori error estimation for the difference scheme on an arbitrary mesh is given. A solution-adaptive algorithm based on the a posteriori error estimation is designed by equidistributing a monitor function. Numerical experiments show that the scheme is second-order uniformly convergent and improves previous results.

This paper presents a numerical approximation technique to solve reaction-diffusion singularly-perturbed differential equation with robin-type boundary conditions. The proposed technique applies cubic splines to discretize the robin-boundary conditions and exponential splines to generate the solution of singularly perturbed differential equation at the internal nodes of a layer adapted grid. The layer adapted grid is generated by equidistributing a positive monitor function. The error estimates indicate that the proposed technique is parameter-uniform second-order convergent and is numerically stable. Numerical experiments have been performed and presented to corroborate the theoretical results.

Moving mesh methods are a widely used approach in the numerical solution of PDEs where the original PDEs are transformed from a physical domain to a computational domain. The objective is to utilize a uniform mesh in the computational domain to get a non-uniform physical mesh that better captures the behavior of the solution. The movement of the physical mesh points can be governed by a moving mesh PDE associated with a corresponding monitor function and both the original PDEs and the moving mesh PDEs must be solved simultaneously. The motivation for this paper is to study a balanced moving mesh method, where the aim is to strike a balance between the properties of the solution of the original PDE and that of the moving mesh PDE. We focus on particular choices of the monitor function that give both a well-behaved mesh transformation and a well-behaved solution in the computational domain. Both theoretical analysis and numerical experiments are presented as illustrations.

Mesh deformation methods are a versatile strategy for solving partial differential equations (PDEs) with a vast variety of practical applications. However, these methods break down for elliptic PDEs with discontinuous coefficients, namely, elliptic interface problems. For this class of problems, the additional interface jump conditions are required to maintain the well-posedness of the governing equation. Consequently, in order to achieve high accuracy and high order convergence, additional numerical algorithms are required to enforce the interface jump conditions in solving elliptic interface problems. The present work introduces an interface technique based adaptively deformed mesh strategy for resolving elliptic interface problems. We take the advantages of the high accuracy, flexibility and robustness of the matched interface and boundary (MIB) method to construct an adaptively deformed mesh based interface method for elliptic equations with discontinuous coefficients. The proposed method generates deformed meshes in the physical domain and solves the transformed governed equations in the computational domain, which maintains regular Cartesian meshes. The mesh deformation is realized by a mesh transformation PDE, which controls the mesh redistribution by a source term. The source term consists of a monitor function, which builds in mesh contraction rules. Both interface geometry based deformed meshes and solution gradient based deformed meshes are constructed to reduce the L(∞) and L(2) errors in solving elliptic interface problems. The proposed adaptively deformed mesh based interface method is extensively validated by many numerical experiments. Numerical results indicate that the adaptively deformed mesh based interface method outperforms the original MIB method for dealing with elliptic interface problems.

An adaptive structured mesh redistribution method (ASMRM) that permits smooth transition from non-uniformly distributed boundary points to solution-adaptive interior points and enables the resolution of complex flow in the complex boundary region as well as away from the boundary is proposed. It is a variant of the traditional variational technique. It involves a combination of static and dynamic monitor functions, the former for mesh distribution in the vicinity of a complex boundary and the latter for mesh adaption with the evolving solution elsewhere. Its effectiveness is demonstrated on some example problems, and it is then applied to a chevron nozzle. The proposed method is shown to be capable of generating a mesh with a good balance of orthogonality and smoothness in the entire domain.

A diffuse interface type model, using an energy-based variational formulation with a free energy that is a function of the density and its gradients is presented. All of the boundary terms are retained and related to external surface forces, which can be of particular interest when considering the fluid–fluid–solid region. The numerical solution of these types of problems can be troublesome if a thin transition layer is desired. Here, Chebyshev pseudospectral methods with mesh adaptation for the solution of diffuse interface type problems are studied. A mesh adaptation algorithm based in the equidistribution principle following a continuation process is derived. In order to achieve high precision for problems exhibiting thin transition layers, a modified version of the arc-length monitor function is proposed which yields a sufficiently smooth coordinate transformation. At every step of the continuation process, a fixed number of iterations is implemented, so that the equidistribution equations are not solved completely at each step, which saves a considerable amount of computational effort. Numerical results for the static phase field model exhibiting thin transition layers are presented.

This paper presents a method of degradation assessment for electronic equipments and its integration within the avionics maintenance architecture. The health monitoring function is based on physics of failure approach. At first, we explain the current centralized maintenance concept, and then we present a description of the built in health monitor function integration at board level, and the communication process with the maintenance data centralizer.

We consider the problem of computing blow-up solutions of chemotaxis systems, or the so-called chemotactic collapse. In two spatial dimensions, such solutions can have approximate self-similar behaviour, which can be very challenging to verify in numerical simulations [cf. Betterton and Brenner, Collapsing bacterial cylinders, Phys. Rev. E 64 (2001) 061904]. We analyse a dynamic (scale-invariant) remeshing method which performs spatial mesh movement based upon equidistribution. Using a suitably chosen monitor function, the numerical solution resolves the fine detail in the asymptotic solution structure, such that the computations are seen to be fully consistent with the asymptotic description of the collapse phenomenon given by Herrero and Velazquez [Singularity patterns in a chemotaxis model, Math. Ann. 306 (1996) 583-623]. We believe that the methods we construct are ideally suited to a large number of problems in mathematical biology for which collapse phenomena are expected.

An r -adaptive finite-element method based on moving-mesh partial differential equations (PDEs) and an error indicator is presented. The error indicator is obtained by applying a technique developed by Bank and Weiser to elliptic equations which result in this case from temporal discretization of the underlying physical PDEs on moving meshes. The construction of the monitor function based on the error indicator is discussed. Numerical results obtained with the current method and the commonly used method based on solution gradients are presented and analyzed for several examples. c ∞ 2001 Academic Press

Numerical experiments are described that illustrate some important features of the performance of moving mesh methods for solving one-dimensional partial differential equations (PDEs). The particular method considered here is an adaptive finite difference method based on the equidistribution of a monitor function and it is one of the moving mesh methods proposed by W. Huang, Y. Ren, and R. D. Russell (1994, SIAM J. Numer. Anal.31 709). We show how the accuracy of the computations is strongly dependent on the choice of monitor function, and we present a monitor function that yields an optimal rate of convergence. Motivated by efficiency considerations for problems in two or more space dimensions, we demonstrate a robust and efficient algorithm in which the mesh equations are uncoupled from the physical PDE. The accuracy and efficiency of the various formulations of the algorithm are considered and a novel automatic time-step control mechanism is integrated into the scheme.

The paper investigates the viability of using moving mesh methods to simulate travelling wave solutions of Fisher's equation. Results are presented that illustrate the weaknesses in moving mesh methods based on equidistribution of some popular monitor functions. It is shown that knowledge of the differential equation and the travelling wave solution may be used to construct a monitor function that yields accurate results with suitably chosen moving mesh methods. A comparison is made between a moving mesh partial differential equation and a moving mesh differential-algebraic equation for the evolution in time.