The theory of density representation of finite element meshes. Examples of density operators with quadrilateral elements in the mapped domain

Hugger, Jens

doi:10.1016/0045-7825(93)90223-k

Cited by 15 publications

(6 citation statements)

References 8 publications

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“…By forming K~IY~'YK~P~YI'Y, (53) is easily found in the simple case with constant densities. For the general case, the proofs and exact conditions under which the results hold are similar to those of the square element case stated in Section I1 and proved in [8].…”

Section: 'mentioning

confidence: 76%

“…The algorithm presented in [8] for the recovery of meshes from a density and a number of elements, such that ( 3 ) is satisfied, is the following.…”

Section: (4)mentioning

confidence: 99%

“…. 8 The square itself is a compatible mesh. r f T is a compatible mesh, then T' obtained by dividing any element of T into 2 congruent rectangles is also compatible.…”

Section: Definition 2 (Compatible Meshes)mentioning

confidence: 99%

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Recovery of the optimal mesh density function and near‐optimal meshes with rectangular elements in the mapped domain

Hugger

1995

Numerical Methods Partial

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A method is developed for the recovery and representation of near-optimal finite-element meshes of rectangular elements in the mapped domain. The method extends a previously known method, which was restricted to square elements in the mapped domain. The method was developed to obtain a simple, cost-effective representation of computational meshes for adaptive finite-element methods solving parametrized problems. 0 1995 John Wiley & Sons, Inc.

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Section: 'mentioning

confidence: 76%

“…The algorithm presented in [8] for the recovery of meshes from a density and a number of elements, such that ( 3 ) is satisfied, is the following.…”

Section: (4)mentioning

confidence: 99%

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Recovery of the optimal mesh density function and near‐optimal meshes with rectangular elements in the mapped domain

Hugger

1995

Numerical Methods Partial

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“…A mesh density function [33], [34] should be a nonnegative function : [ , ] → ℜ + , , ∈ ℜ that satisfies ∫︀ ( ) = 1 and equals to zero at countably points. Since any nonnegative function : [ , ] → ℜ + that has only countably many zeros can be summarised as:…”

Section: A Density Functionmentioning

confidence: 99%

An Interactive Fuzzy Physical Programming for Solving Multiobjective Skip Entry Problem

Chai

Tsourdos

Xia

2017

IEEE Trans. Aerosp. Electron. Syst.

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The multi-criteria trajectory planning for Space Manoeuvre Vehicle (SMV) is recognised as a challenging problem. Because of the nonlinearity and uncertainty in the dynamic model and even the objectives, it is hard for decision makers to balance all of the preference indices without violating strict path and box constraints. In this paper, to provide the designer an effective method and solve the trajectory hopping problem, an Interactive Fuzzy Physical Programming (IFPP) algorithm is introduced. A new multi-objective SMV optimal control problem is formulated and parameterized using an adaptive technique. By using the density function, the oscillations of the trajectory can be captured effectively. In addition, an interactive decision-making strategy is applied to modify the current designer's preferences during optimization process. Two realistic decision-making scenarios are conducted by using the proposed algorithm; Simulation results indicated that without driving objective functions out of the tolerable region, the proposed approach can have better performance in terms of the satisfactory degree compared with other approaches like traditional weighted-sum method, Goal Programming (GP) and fuzzy goal programming (FGP). Also, the results can satisfy the current preferences given by the decision makers. Therefore, The method is potentially feasible for solving multi-criteria SMV trajectory planning problems.

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“…Mesh generation and adaptation is a common topic in many areas of engineering and applied mathematics. The notion of mesh density function for mesh generation and refinement has been used in the finite element method field [5,6]. The concept of density functions is also similar to monitor functions used for the numerical solution of partial differential equations [7].…”

Section: Introductionmentioning

confidence: 99%

Density Functions for Mesh Refinement in Numerical Optimal Control

Zhao

Tsiotras

2011

Journal of Guidance, Control, and Dynamics

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This paper introduces an efficient and simple method for mesh point distribution for solving optimal control problems using direct methods. The method is based on density (or monitor) functions, which have been used extensively for mesh refinement in other areas such as partial differential equations and finite element methods. Subsequently, the problem of mesh refinement is converted to a problem of finding an appropriate density function. It is shown that an appropriate choice of density function may help increase the accuracy of the solution and improve numerical robustness.

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The theory of density representation of finite element meshes. Examples of density operators with quadrilateral elements in the mapped domain

Cited by 15 publications

References 8 publications

Recovery of the optimal mesh density function and near‐optimal meshes with rectangular elements in the mapped domain

Recovery of the optimal mesh density function and near‐optimal meshes with rectangular elements in the mapped domain

An Interactive Fuzzy Physical Programming for Solving Multiobjective Skip Entry Problem

Density Functions for Mesh Refinement in Numerical Optimal Control

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