On the mesh nonsingularity of the moving mesh PDE method

Huang, Weizhang; Kamenski, Lennard

doi:10.1090/mcom/3271

Cited by 45 publications

(51 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof. The proof is very much the same as that for [15,Theorem 4.3] for the bulk mesh case. The key ideas to the proof are the monotonicity and boundedness of I h (T h (t)) and the compactness of S. With these holding for the surface mesh case, one can readily prove the three properties.…”

Section: Mesh Nonsingularitymentioning

confidence: 66%

A surface moving mesh method based on equidistribution and alignment

Kolasinski¹,

Huang²

2020

Journal of Computational Physics

Self Cite

View full text Add to dashboard Cite

A surface moving mesh method is presented for general surfaces with or without explicit parameterization. The method can be viewed as a nontrivial extension of the moving mesh partial differential equation method that has been developed for bulk meshes and demonstrated to work well for various applications. The main challenges in the development of surface mesh movement come from the fact that the Jacobian matrix of the affine mapping between the reference element and any simplicial surface element is not square. The development starts with revealing the relation between the area of a surface element in the Euclidean or Riemannian metric and the Jacobian matrix of the corresponding affine mapping, formulating the equidistribution and alignment conditions for surface meshes, and establishing a meshing energy function based on the conditions. The moving mesh equation is then defined as the gradient system of the energy function, with the nodal mesh velocities being projected onto the underlying surface. The analytical expression for the mesh velocities is obtained in a compact, matrix form, which makes the implementation of the new method on a computer relatively easy and robust. Moreover, it is analytically shown that any mesh trajectory generated by the method remains nonsingular if it is so initially. It is emphasized that the method is developed directly on surface meshes, making no use of any information on surface parameterization. It utilizes surface normal vectors to ensure that the mesh vertices remain on the surface while moving, and also assumes that the initial surface mesh is given. The new method can apply to general surfaces with or without explicit parameterization since the surface normal vectors can be computed even when the surface only has a numerical representation. A selection of twoand three-dimensional examples are presented.

show abstract

Section: Mesh Nonsingularitymentioning

confidence: 66%

A surface moving mesh method based on equidistribution and alignment

Kolasinski¹,

Huang²

2020

Journal of Computational Physics

Self Cite

View full text Add to dashboard Cite

show abstract

“…The next two lemmas establish bounds for A jj with a more explicit geometric interpretation than (14). For this, we denote the diameter and the minimal height of K in the metric D −1 K by h K,D −1 and a K,D −1 , respectively.…”

Section: Preliminary Estimates On the Extreme Eigenvalues Of The Massmentioning

confidence: 98%

Conditioning of implicit Runge–Kutta integration for finite element approximation of linear diffusion equations on anisotropic meshes

Huang

Kamenski²,

Lang

2021

Journal of Computational and Applied Mathematics

Self Cite

View full text Add to dashboard Cite

The conditioning of implicit Runge-Kutta (RK) integration for linear finite element approximation of diffusion equations on general anisotropic meshes is investigated. Bounds are established for the condition number of the resulting linear system with and without diagonal preconditioning for the implicit Euler (the simplest implicit RK method) and general implicit RK methods. Two solution strategies are considered for the linear system resulting from general implicit RK integration: the simultaneous solution where the system is solved as a whole and a successive solution which follows the commonly used implementation of implicit RK methods to first transform the system into a number of smaller systems using the Jordan normal form of the RK matrix and then solve them successively.For the simultaneous solution in case of a positive semidefinite symmetric part of the RK coefficient matrix and for the successive solution it is shown that the conditioning of an implicit RK method behaves like that of the implicit Euler method. If the smallest eigenvalue of the symmetric part of the RK coefficient matrix is negative and the simultaneous solution strategy is used, an upper bound on the time step is given so that the system matrix is positive definite.The obtained bounds for the condition number have explicit geometric interpretations and take the interplay between the diffusion matrix and the mesh geometry into full consideration. They show that there are three mesh-dependent factors that can affect the conditioning: the number of elements, the mesh nonuniformity measured in the Euclidean metric, and the mesh nonuniformity with respect to the inverse of the diffusion matrix. They also reveal that the preconditioning using the diagonal of the system matrix, the mass matrix, or the lumped mass matrix can effectively eliminate the effects of the mesh nonuniformity measured in the Euclidean metric. Illustrative numerical examples are given. Lemma 3.2 ([16, Lemma 2.3]). The mass matrix M and the lumped mass matrix M lump are related by

show abstract

“…where ∂I h ∂xi is a row vector, P i is a positive function chosen as P i = det(M(x i )) 1 4 (to make (20) to be invariant under the scaling transformation of M), and τ > 0 is a positive parameter used to adjust the response time of mesh movement to the changes in M. It has been proven in [23] that the mesh governed by (20) stays non-singular if it is non-singular initially. This result holds for any convex or concave domain in any dimension and for the semi-discrete form (20) or a fully-discrete form of (20).…”

Section: Denote the Coordinates Of The Vertices Of T N H And T N+1mentioning

confidence: 99%

Moving mesh simulation of contact sets in two dimensional models of elastic–electrostatic deflection problems

DiPietro

Haynes

Huang

et al. 2018

Journal of Computational Physics

Self Cite

View full text Add to dashboard Cite

Numerical and analytical methods are developed for the investigation of contact sets in electrostatic-elastic deflections modeling micro-electro mechanical systems. The model for the membrane deflection is a fourth-order semi-linear partial differential equation and the contact events occur in this system as finite time singularities. Primary research interest is in the dependence of the contact set on model parameters and the geometry of the domain. An adaptive numerical strategy is developed based on a moving mesh partial differential equation to dynamically relocate a fixed number of mesh points to increase density where the solution has fine scale detail, particularly in the vicinity of forming singularities. To complement this computational tool, a singular perturbation analysis is used to develop a geometric theory for predicting the possible contact sets. The validity of these two approaches are demonstrated with a variety of test cases.

show abstract

On the mesh nonsingularity of the moving mesh PDE method

Cited by 45 publications

References 28 publications

A surface moving mesh method based on equidistribution and alignment

A surface moving mesh method based on equidistribution and alignment

Conditioning of implicit Runge–Kutta integration for finite element approximation of linear diffusion equations on anisotropic meshes

Moving mesh simulation of contact sets in two dimensional models of elastic–electrostatic deflection problems

Contact Info

Product

Resources

About