A geometric discretization and a simple implementation for variational mesh generation and adaptation

Huang, Weizhang; Kamenski, Lennard

doi:10.1016/j.jcp.2015.08.032

Cited by 53 publications

(88 citation statements)

References 22 publications

(53 reference statements)

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“…It is straightforward to solve for x j , j = 1, …, N v directly for a given reference computational mesh

{scriptT}_{c}

. The formulas for this so‐called x ‐formulation of the MMPDE method are given in [, eqs. (39)–(41)] but omitted here to save space.…”

Section: The Moving Mesh Femmentioning

confidence: 99%

“…Then the MMPDE for the computational vertices is defined as the gradient system of I h as

\frac{d {bold-italicξ}_{j}}{italicdt} = - \frac{P_{j}}{τ} {([], \frac{\partial I_{h}}{\partial {bold-italicξ}_{j}})}^{T}, j = 1, \dots, N_{v},

where the row vector ∂I h / ∂ ξ j is the derivative of I h with respect to ξ j , τ > 0 is a parameter used to adjust how fast the mesh movement reacts to any change in the metric tensor, and

P_{j} = \det {((), double-struckM ((), {bold-italicx}_{j}))}^{\frac{p - 1}{2}}

is chosen such that is invariant under the scaling transformation of

M

M \to c M

for any positive constant c . The derivative ∂I h / ∂ ξ j can be found analytically using the notion of scalar‐by‐matrix differentiation . Using this, we can rewrite into

\frac{d {bold-italicξ}_{j}}{italicdt} = \frac{P_{j}}{τ} \underset{K \in ω_{j}}{false\sum} ∣ K ∣ {bold-italicv}_{j_{K}}^{K}, j = 1, \dots, N_{v},

where ω j denotes the element patch associated with the vertex x j , j K is the local index of the same vertex on the element K , and

{bold-italicv}_{j_{K}}^{K}

is the velocity contributed by K to the vertex x j .…”

Section: The Moving Mesh Femmentioning

confidence: 99%

See 1 more Smart Citation

Adaptive finite element solution of the porous medium equation in pressure formulation

Ngo

Huang

2019

Numerical Methods Partial

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The pressure formulation of the porous medium equation has been commonly used in theoretical studies due to its much better regularities than the original formulation. The goal here is to study its use in the adaptive moving mesh finite element solution. The free boundary is traced explicitly through Darcy's law. The method is shown numerically second‐order in space and first‐order in time in the pressure variable. Moreover, the convergence order of the error in the location of the free boundary is almost second‐order in the maximum norm. However, numerical results also show that the convergence order in the original variable stays between first‐order and second‐order in L1 norm or between 0.5th‐order and first‐order in L2 norm. Nevertheless, the current method can offer some advantages over numerical methods based on the original formulation for situations with large exponents or when a more accurate location of the free boundary is desired.

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“…It is straightforward to solve for x j , j = 1, …, N v directly for a given reference computational mesh

{scriptT}_{c}

. The formulas for this so‐called x ‐formulation of the MMPDE method are given in [, eqs. (39)–(41)] but omitted here to save space.…”

Section: The Moving Mesh Femmentioning

confidence: 99%

“…Then the MMPDE for the computational vertices is defined as the gradient system of I h as

\frac{d {bold-italicξ}_{j}}{italicdt} = - \frac{P_{j}}{τ} {([], \frac{\partial I_{h}}{\partial {bold-italicξ}_{j}})}^{T}, j = 1, \dots, N_{v},

where the row vector ∂I h / ∂ ξ j is the derivative of I h with respect to ξ j , τ > 0 is a parameter used to adjust how fast the mesh movement reacts to any change in the metric tensor, and

P_{j} = \det {((), double-struckM ((), {bold-italicx}_{j}))}^{\frac{p - 1}{2}}

is chosen such that is invariant under the scaling transformation of

M

M \to c M

for any positive constant c . The derivative ∂I h / ∂ ξ j can be found analytically using the notion of scalar‐by‐matrix differentiation . Using this, we can rewrite into

\frac{d {bold-italicξ}_{j}}{italicdt} = \frac{P_{j}}{τ} \underset{K \in ω_{j}}{false\sum} ∣ K ∣ {bold-italicv}_{j_{K}}^{K}, j = 1, \dots, N_{v},

where ω j denotes the element patch associated with the vertex x j , j K is the local index of the same vertex on the element K , and

{bold-italicv}_{j_{K}}^{K}

is the velocity contributed by K to the vertex x j .…”

Section: The Moving Mesh Femmentioning

confidence: 99%

Adaptive finite element solution of the porous medium equation in pressure formulation

Ngo

Huang

2019

Numerical Methods Partial

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“…When a metric tensor is used, the above formulation needs to be modified to include the metric tensor; see Ref. [14] for related formulation.…”

Section: A the Moving Mesh Techniquementioning

confidence: 99%

Guidance of multi-agent fixed-wing aircraft using a moving mesh method

Kim

Keshmiri

Huang

et al. 2015

2015 International Conference on Unmanned Aircraft Systems (ICUAS)

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This paper presents implementation of a novel guidance logic for multi-agent fixed-wing unmanned aerial systems using a moving mesh method. The moving mesh method is originally designed for use in the adaptive numerical solution of partial differential equations where a high proportion of mesh points are placed in the regions of large solution variation and few points in the rest of the domain. In this work, the positions of the aircraft are considered as mesh nodes and connected to form a triangular mesh in two spatial dimensions. The outer aircraft, or the boundary nodes of the mesh, are moved with the velocities which are specified using the the relative distance from and the heading angle of a reference point while keeping the formation of the aircraft in a desired shape. The inner agents or the interior mesh nodes, on the other hand, are moved with a moving mesh technique to keep the whole mesh as uniform as possible. The moving mesh technique has builtin mechanisms to keep the mesh as uniform as possible and prevent nodes from crossing over or tangling. This property can be seen as an automatic collision avoidance mechanism. It also has explicit formulas for nodal velocities, which makes the technique easy to implement on computer. The centralized moving mesh guidance was complimented by a decentralized nonlinear predictive controller to control each aircraft. The mesh nodes are replaced by unmanned aerial systems with nonlinear six degrees of freedom dynamics. To increase flexibility of aircraft in the formation, the moving point concept is used to follow arbitrary linear or curvature shaped flight segments.

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“…This method was applied to approximate the anisotropic diffusion-convection-reaction problems to satisfy a discrete maximum principle, and it was adopted to study the porous medium equation and approximate the solution [36]. For more information about the moving mesh method, please refer to [37][38][39][40][41].…”

Section: Introductionmentioning

confidence: 99%

Finite Element Method of BBM-Burgers Equation with Dissipative Term Based on Adaptive Moving Mesh

Gao

et al. 2017

Discrete Dynamics in Nature and Society

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A finite element model is proposed for the Benjamin-Bona-Mahony-Burgers (BBM-Burgers) equation with a high-order dissipative term; the scheme is based on adaptive moving meshes. The model can be applied to the equations with spatial-time mixed derivatives and high-order derivative terms. In this scheme, new variables are needed to make the equation become a coupled system, and then the linear finite element method is used to discretize the spatial derivative and the fifth-order Radau IIA method is used to discretize the time derivative. The simulations of 1D and 2D BBM-Burgers equations with high-order dissipative terms are presented in numerical examples. The numerical results show that the method keeps a second-order convergence in space and provides a smaller error than that based on the fixed mesh, which demonstrates the effectiveness and feasibility of the finite element method based on the moving mesh. We also study the effect of the dissipative terms with different coefficients in the equation; by numerical simulations, we find that the dissipative term plays a more important role than in dissipation.

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A geometric discretization and a simple implementation for variational mesh generation and adaptation

Cited by 53 publications

References 22 publications

Adaptive finite element solution of the porous medium equation in pressure formulation

Adaptive finite element solution of the porous medium equation in pressure formulation

Guidance of multi-agent fixed-wing aircraft using a moving mesh method

Finite Element Method of BBM-Burgers Equation with Dissipative Term Based on Adaptive Moving Mesh

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