A Space-Time Adaptive Algorithm to Illustrate the Lack of Collision of a Rigid Disk Falling in an Incompressible Fluid

Samuel Dubuis; Marco Picasso; Peter Wittwer

doi:10.1515/cmam-2020-0046

Article

A Space-Time Adaptive Algorithm to Illustrate the Lack of Collision of a Rigid Disk Falling in an Incompressible Fluid

Samuel Dubuis , Marco Picasso and Peter Wittwer

Published/Copyright: November 7, 2020

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Computational Methods in Applied Mathematics Volume 21 Issue 2

Abstract

A space-time adaptive algorithm to solve the motion of a rigid disk in an incompressible Newtonian fluid is presented, which allows collision or quasi-collision processes to be computed with high accuracy. In particular, we recover the theoretical result proven in [M. Hillairet, Lack of collision between solid bodies in a 2D incompressible viscous flow, Comm. Partial Differential Equations 32 2007, 7–9, 1345–1371], that the disk will never touch the boundary of the domain in finite time. Anisotropic, continuous piecewise linear finite elements are used for the space discretization, the Euler scheme for the time discretization. The adaptive criteria are based on a posteriori error estimates for simpler problems.

Keywords: Anisotropic Finite Elements; Adaptive Method; Solid-Fluid Interactions

MSC 2010: 65M50; 74F10

A Derivation of the Penalty Method (2.1)

We present the derivation of the penalty formulation (2.1) used to approximate the motion of the rigid disk inside the cavity. We start from the physical model presented in [12]. We study the motion of a rigid disk of radius R inside a bounded, convex cavity $Ω ⊂ ℝ 2$ filled with a fluid of constant viscosity and density. The fluid is governed by the incompressible Navier–Stokes equations while the dynamic of the rigid disk is ruled by the Newton law. Given a final time $T ∈ ( 0 , + ∞ ]$ , we denote by $ℬ ⁢ ( 𝑿 ⁢ ( t ) )$ the disk centered at $𝑿 ⁢ ( t )$ , where $t ∈ [ 0 , T ]$ . The equations of motions reads [12]: find $( 𝑿 , 𝐮 , p , 𝑽 , ω )$ the solutions of

(A.1)

${ ρ ℱ ⁢ ( ∂ ⁡ 𝐮 ∂ ⁡ t + ( 𝐮 ⋅ ∇ ) ⁢ 𝐮 ) - μ ℱ ⁢ Δ ⁢ 𝐮 + ∇ ⁡ p = ρ ℱ ⁢ 𝐠 , ( 𝐱 , t ) ∈ Ω ∖ ℬ ⁢ ( 𝑿 ⁢ ( t ) ) ¯ × ( 0 , T ] , div ⁡ 𝐮 = 0 , ( 𝐱 , t ) ∈ Ω ∖ ℬ ⁢ ( 𝑿 ⁢ ( t ) ) ¯ × ( 0 , T ] , 𝐮 = 0 , ( 𝐱 , t ) ∈ ∂ ⁡ Ω × ( 0 , T ] , 𝐮 ⁢ ( 𝐱 , t ) = 𝑽 ⁢ ( t ) + ω ⁢ ( 𝐱 - 𝑿 ⁢ ( t ) ) ⊥ , ( 𝐱 , t ) ∈ ∂ ⁡ ℬ ⁢ ( 𝑿 ⁢ ( t ) ) × ( 0 , T ] , 𝑿 ˙ ⁢ ( t ) = 𝑽 ⁢ ( t ) ,$

together with

(A.2)

${ m ⁢ 𝐠 - ∫ ∂ ⁡ ℬ ⁢ ( 𝑿 ⁢ ( t ) ) ( 2 ⁢ μ ℱ ⁢ D ⁢ ( 𝐮 ) - p ⁢ I ) ⁢ 𝒏 ⁢ 𝑑 σ = m ⁢ 𝑽 ˙ , t ∈ ( 0 , T ] , ∫ ℬ ⁢ ( 𝑿 ⁢ ( t ) ) ρ ℬ ⁢ 𝐠 ⋅ ( 𝐱 - 𝑿 ⁢ ( t ) ) ⊥ - ∫ ∂ ⁡ ℬ ⁢ ( 𝑿 ⁢ ( t ) ) ( 2 ⁢ μ ℱ ⁢ D ⁢ ( 𝐮 ) - p ⁢ I ) ⁢ ( 𝐱 - 𝑿 ⁢ ( t ) ) ⊥ ⋅ 𝒏 ⁢ 𝑑 σ = J ⁢ ω ˙ , t ∈ ( 0 , T ] ,$

where $𝐠$ is gravitational acceleration, $μ ℱ$ and $ρ ℱ$ denote the viscosity and the density of the fluid, $ρ ℬ$ the density of the disk, m and J its mass and inertia. In the above system, $( 𝐮 , p )$ are the velocity and pressure fields of the fluid and $( 𝑽 , ω )$ the translational and rotational speeds of $ℬ ⁢ ( 𝑿 ⁢ ( t ) )$ . Here above we note

$( 𝐱 - 𝑿 ⁢ ( t ) ) ⊥ = ( - ( x 2 - X 2 ⁢ ( t ) ) , x 1 - X 1 ⁢ ( t ) ) ∈ ℝ 2 .$

Note that since $𝐠$ is applied at the center of mass, its angular momentum satisfies in particular

$∫ ℬ ⁢ ( 𝑿 ⁢ ( t ) ) ρ ℬ ⁢ 𝐠 ⋅ ( 𝐱 - 𝑿 ⁢ ( t ) ) ⊥ = 0$

which is also true for any constant vector $𝑽$ inside the disk, i.e.,

$∫ ℬ ⁢ ( 𝑿 ⁢ ( t ) ) 𝑽 ⋅ ( 𝐱 - 𝑿 ⁢ ( t ) ) ⊥ = 0 .$

To derive equations (2.1), the first step consists to establish a variational formulation that can easily be approximated with Galerkin methods, that is to say involving test functions that are defined on the whole cavity Ω. We follow the formulation introduced in [15]. Let us consider the problem at a given time t, and to lighten the presentation, we drop the dependence of $𝑿$ upon the time. To extend the velocity field inside the disk, one may consider the space of rigid motion inside $ℬ ⁢ ( 𝑿 )$ given by

$H ℬ ⁢ ( 𝑿 ) = { 𝐯 ∈ ( H 0 1 ⁢ ( Ω ) ) 2 : there exists ⁢ ( 𝑽 , ω ) ∈ ℝ 2 × ℝ ⁢ such that ⁢ 𝐯 = 𝑽 + ω ⁢ ( 𝐱 - 𝑿 ) ⊥ ⁢ a.e. in ⁢ ℬ ⁢ ( 𝑿 ) }$

that has the equivalent definition (see for instance [27])

$H ℬ ⁢ ( 𝑿 ) = { 𝐯 ∈ ( H 0 1 ⁢ ( Ω ) ) 2 : D ⁢ ( 𝐯 ) = 0 ⁢ a.e. in ⁢ ℬ ⁢ ( 𝑿 ) } .$

We now multiply the momentum equation of (A.1) by a test function $𝐯 ∈ H ℬ ⁢ ( 𝑿 )$ and using the fact that for free divergence field

$Δ ⁢ 𝐮 = 2 ⁢ div ⁡ ( D ⁢ ( 𝐮 ) ) ,$

one has after integration by part

$∫ Ω ∖ ℬ ⁢ ( 𝑿 ) ¯ ρ ℱ ⁢ ( ∂ ⁡ 𝐮 ∂ ⁡ t + 𝐮 ⋅ ∇ ⁡ 𝐮 ) ⋅ 𝐯 + 2 ⁢ μ ℱ ⁢ ∫ Ω ∖ ℬ ⁢ ( 𝑿 ) ¯ D ⁢ ( 𝐮 ) : D ⁢ ( 𝐯 ) - ∫ Ω ∖ ℬ ⁢ ( 𝑿 ) ¯ p ⁢ div ⁡ 𝐯 - ∫ ∂ ⁡ ℬ ⁢ ( 𝑿 ) ( 2 ⁢ μ ℱ ⁢ D ⁢ ( 𝐮 ) - p ⁢ I ) ⁢ 𝒏 ⋅ 𝐯 = ∫ Ω ∖ ℬ ⁢ ( 𝑿 ) ¯ ρ ℱ ⁢ 𝐠 ⋅ 𝐯 .$

Using the Newton’s laws (A.2) and the fact that $𝐯 = 𝑽 v + ω v ⁢ ( 𝐱 - 𝑿 ) ⊥$ in $ℬ ⁢ ( 𝑿 )$ , one can write that

$- ∫ ∂ ⁡ ℬ ⁢ ( 𝑿 ) ( 2 ⁢ μ ℱ ⁢ D ⁢ ( 𝐮 ) - p ⁢ I ) ⁢ 𝒏 ⋅ 𝐯 = m ⁢ 𝑽 ˙ ⋅ 𝑽 v - m ⁢ 𝐠 ⋅ 𝑽 v + J ⁢ ω ˙ ⁢ ω v - ∫ ℬ ⁢ ( 𝑿 ) ρ ℬ ⁢ 𝐠 ⋅ ω v ⁢ ( 𝐱 - 𝑿 ) ⊥ .$

Moreover, one has that

$m ⁢ 𝑽 ˙ ⋅ 𝑽 v - m ⁢ 𝐠 ⋅ 𝑽 v = ∫ B ⁢ ( 𝑿 ) ρ ℬ ⁢ 𝑽 ˙ ⋅ 𝑽 v - ∫ B ⁢ ( 𝑿 ) ρ ℬ ⁢ 𝐠 ⋅ 𝑽 v ,$

and

$J ⁢ ω ˙ ⁢ ω v = ∫ B ⁢ ( 𝑿 ) ρ ℬ ⁢ ω ˙ ⁢ ω v ⁢ | 𝐱 - 𝑿 | 2 = ∫ B ⁢ ( 𝑿 ) ρ ℬ ⁢ ω ˙ ⁢ ( 𝐱 - 𝑿 ) ⊥ ⋅ ω v ⁢ ( 𝐱 - 𝑿 ) ⊥ .$

Thus since $𝐮 , 𝐯 ∈ H ℬ ⁢ ( 𝑿 ) ,$ we can prove that

$- ∫ ∂ ⁡ ℬ ⁢ ( 𝑿 ) ( 2 ⁢ μ ℱ ⁢ D ⁢ ( 𝐮 ) - p ⁢ I ) ⁢ 𝒏 ⋅ 𝐯 = m ⁢ 𝑽 ˙ ⋅ 𝑽 v + J ⁢ ω ˙ ⁢ ω v - m ⁢ 𝐠 ⋅ 𝑽 v - ∫ ℬ ⁢ ( 𝑿 ) ρ ℬ ⁢ 𝐠 ⋅ ω v ⁢ ( 𝐱 - 𝑿 ) ⊥$

$= ∫ B ⁢ ( 𝑿 ) ρ ℬ ⁢ ( 𝑽 ˙ ⋅ 𝑽 v + ω ˙ ⁢ ( 𝐱 - 𝑿 ) ⊥ ⋅ ω v ⁢ ( 𝐱 - 𝑿 ) ⊥ )$

$- ∫ B ⁢ ( 𝑿 ) ρ ℬ ⁢ 𝐠 ⋅ 𝑽 v - ∫ ℬ ⁢ ( 𝑿 ) ρ ℬ ⁢ 𝐠 ⋅ ω v ⁢ ( 𝐱 - 𝑿 ) ⊥$

$= ∫ B ⁢ ( 𝑿 ) ρ B ⁢ ( ∂ ⁡ 𝐮 ∂ ⁡ t + ( 𝐮 ⋅ ∇ ) ⁢ 𝐮 ) ⋅ 𝐯 - ∫ B ⁢ ( 𝑿 ) ρ ℬ ⁢ 𝐠 ⋅ 𝐯 ,$

which leads to

$∫ Ω ∖ ℬ ⁢ ( 𝑿 ) ¯ ρ ℱ ⁢ ( ∂ ⁡ 𝐮 ∂ ⁡ t + ( 𝐮 ⋅ ∇ ) ⁢ 𝐮 ) ⋅ 𝐯 + ∫ B ⁢ ( 𝑿 ) ρ B ⁢ ( ∂ ⁡ 𝐮 ∂ ⁡ t + ( 𝐮 ⋅ ∇ ) ⁢ 𝐮 ) ⋅ 𝐯 + 2 ⁢ μ ℱ ⁢ ∫ Ω ∖ ℬ ⁢ ( 𝑿 ) ¯ D ⁢ ( 𝐮 ) : D ⁢ ( 𝐯 ) - ∫ Ω ∖ ℬ ⁢ ( 𝑿 ) ¯ p ⁢ div ⁡ 𝐯$

$= ∫ Ω ∖ ℬ ⁢ ( 𝑿 ) ¯ ρ ℱ ⁢ 𝐠 ⋅ 𝐯 + ∫ B ⁢ ( 𝑿 ) ρ ℬ ⁢ 𝐠 ⋅ 𝐯 .$

Finally, since $D ⁢ ( 𝐮 ) | ℬ ⁢ ( 𝑿 ) = D ⁢ ( 𝐯 ) | ℬ ⁢ ( 𝑿 ) = 0$ (in particular $div ⁡ 𝐮 = div ⁡ 𝐯 = 0$ inside the disk), it is possible to write the previous equation on the whole domain Ω, which leads to the variational formulation

$∫ Ω ρ 𝑿 ⁢ ( ∂ ⁡ 𝐮 ∂ ⁡ t + 𝐮 ⋅ ∇ ⁡ 𝐮 ) ⋅ 𝐯 + 2 ⁢ μ ℱ ⁢ ∫ Ω D ⁢ ( 𝐮 ) : D ⁢ ( 𝐯 ) - ∫ Ω p ⁢ div ⁡ 𝐯 = ∫ Ω ρ 𝑿 ⁢ 𝐠 ⋅ 𝐯$

$for all ⁢ 𝐯 ∈ H ℬ ⁢ ( 𝑿 ) ,$

$∫ Ω q ⁢ div ⁡ 𝐮 = 0$

$for all ⁢ q ∈ L 2 ⁢ ( Ω ) ,$

where we note

$ρ 𝑿 = ρ ℱ ⁢ χ Ω ∖ ℬ ⁢ ( 𝑿 ) ¯ + ρ ℬ ⁢ χ ℬ ⁢ ( 𝑿 ) .$

The constrains $D ⁢ ( 𝐮 ) B ⁢ ( 𝑿 ) = D ⁢ ( 𝐯 ) B ⁢ ( 𝑿 ) = 0$ can be approximated by a penalization of the viscous term. Thus the variational formulation can be written with test functions taken in $H 0 1 ⁢ ( Ω )$ by considering (we keep the same notations for the unknowns to lighten the presentation)

$∫ Ω ρ 𝑿 ⁢ ( ∂ ⁡ 𝐮 ∂ ⁡ t + ( 𝐮 ⋅ ∇ ) ⁢ 𝐮 ) ⋅ 𝐯 + ∫ Ω 2 ⁢ μ 𝑿 ⁢ D ⁢ ( 𝐮 ) : D ⁢ ( 𝐯 ) - ∫ Ω p ⁢ div ⁡ 𝐯 = ∫ Ω ρ 𝑿 ⁢ 𝐠 ⋅ 𝐯 for all ⁢ 𝐯 ∈ ( H 0 1 ⁢ ( Ω ) ) 2 ,$

where

$μ 𝑿 = μ ℱ ⁢ χ Ω ∖ ℬ ⁢ ( 𝑿 ) ¯ + 1 ε ⁢ χ ℬ ⁢ ( 𝑿 ) , ε ≪ 1 .$

Written in the strong form, the approximated variational formulation gives

$ρ 𝑿 ⁢ ( ∂ ⁡ 𝐮 ∂ ⁡ t + ( 𝐮 ⋅ ∇ ) ⁢ 𝐮 ) - div ⁡ ( 2 ⁢ μ 𝑿 ⁢ D ⁢ ( 𝐮 ) ) + ∇ ⁡ p = ρ 𝑿 ⁢ 𝐠$

with $div ⁡ 𝐮 = 0$ . To obtain (2.1), we finally have to consider equations to compute the translational and rotational velocities inside the disk. In our approach, we assume that we are in a symmetric situation and that the disk does not spin. Thus it remains to have an equation to update only the position of the disk that is obtained by approximating the velocity inside the disk by taking its mean value

(A.3)

$𝑿 ˙ = 1 π ⁢ R 2 ⁢ ∫ ℬ ⁢ ( 𝑿 ) 𝐮 .$

Observe that the last equation reduces to $𝑿 ˙ = 𝐮$ in $ℬ ⁢ ( t )$ if $𝐮$ is constant in $ℬ ⁢ ( t )$ , that is expected when $ε → 0$ . Thus it is justified to use (A.3) to approximate the motion of the disk.

B Justification of (2.8)

We now prove an a posteriori error estimate for the time discretization of a simplified problem. We simplify the first equation of (2.1) and consider the following problem:

(B.1)

${ - div ⁡ ( 2 ⁢ μ 𝑿 ⁢ D ⁢ ( 𝐮 𝑿 ) ) + ∇ ⁡ p 𝑿 = ρ 𝑿 ⁢ 𝐠 in ⁢ Ω × ( 0 , T ) , div ⁡ 𝐮 𝑿 = 0 in ⁢ Ω × ( 0 , T ) , 𝑿 ˙ ⁢ ( t ) = 1 π ⁢ R 2 ⁢ ∫ ℬ ⁢ ( 𝑿 ⁢ ( t ) ) 𝐮 𝑿 ⁢ ( x , t ) ⁢ 𝑑 x , t ∈ ( 0 , T ) ,$

where $ρ 𝑿$ and $μ 𝑿$ are still defined by (2.2). Here $𝐮 𝑿$ is zero on the boundary $∂ ⁡ Ω$ and $𝑿 ⁢ ( 0 ) = 𝑿 0$ .

We consider the forward Euler method. Given a integer $N > 0$ and a partition $0 = t 0 < t 1 < … < t N = T$ of $[ 0 , T ]$ , let $τ n + 1 = t n + 1 - t n$ , $τ = max n = 0 , 1 , 2 , … , N - 1 ⁡ τ n + 1$ . For every $n = 0 , 1 , 2 , … , N - 1$ , given $𝑿 n$ , we compute first $( 𝐮 𝑿 n , p 𝑿 n )$ solution of

(B.2)

${ - div ⁡ ( 2 ⁢ μ 𝑿 n ⁢ D ⁢ ( 𝐮 𝑿 n ) ) + ∇ ⁡ p 𝑿 n = ρ 𝑿 n ⁢ 𝐠 in ⁢ Ω , div ⁡ 𝐮 𝑿 n = 0 in ⁢ Ω , 𝐮 𝑿 n = 0 on ⁢ ∂ ⁡ Ω ,$

then $𝑿 n + 1$ solution of

$𝑿 n + 1 - 𝑿 n τ n + 1 = 1 π ⁢ R 2 ⁢ ∫ ℬ ⁢ ( 𝑿 n ) 𝐮 𝑿 n ⁢ 𝑑 x .$

We assume that there exists a unique

$( 𝑿 , 𝐮 𝑿 , p 𝑿 ) ∈ C 1 ⁢ [ 0 , T ] × C ⁢ ( [ 0 , T ] ; ( H 0 1 ⁢ ( Ω ) ) 2 ) × L 2 ⁢ ( ( 0 , T ) ; L 0 2 ⁢ ( Ω ) )$

solution of (B.1), that $d ( ℬ ( 𝑿 ( t ) ; ∂ Ω ) > 0$ for all $t ∈ [ 0 , T ]$ , and that $𝑿 n , 𝐮 𝑿 n , p 𝑿 n$ converge to $𝑿 ⁢ ( t n ) , 𝐮 𝑿 ⁢ ( t n ) , p 𝑿 ⁢ ( t n )$ when τ goes to zero, and prove an a posteriori error estimate.

Observe that problem (B.1) reads: starting from $𝑿 ⁢ ( 0 ) = 𝑿 0$ , find $𝑿 ⁢ ( t )$ , for $t > 0$ , such that

$𝑿 ˙ ⁢ ( t ) = 𝑭 ⁢ ( 𝑿 ⁢ ( t ) ) ,$

where, for any $𝒀 ∈ ℝ 2$ such that $ℬ ⁢ ( 𝒀 )$ is compactly supported in Ω, $𝑭 ⁢ ( 𝒀 )$ is given by

$𝑭 ⁢ ( 𝒀 ) = 1 π ⁢ R 2 ⁢ ∫ ℬ ⁢ ( 𝒀 ) 𝐮 𝒀 ⁢ 𝑑 x ,$

and $( 𝐮 𝒀 , p 𝒀 ) ∈ ( H 0 1 ⁢ ( Ω ) ) 2 × L 0 2 ⁢ ( Ω )$ is the weak solution of

${ - div ⁡ ( 2 ⁢ μ 𝒀 ⁢ D ⁢ ( 𝐮 𝒀 ) ) + ∇ ⁡ p 𝒀 = ρ 𝒀 ⁢ 𝐠 , div ⁡ 𝐮 𝒀 = 0 ,$

where $μ 𝒀 , ρ 𝒀$ are defined as in (2.2). In order to prove an a posteriori error estimate, we need to prove that $𝑭$ is Lipschitz.

Lemma 1.

Let $Y = ( Y 1 , Y 2 ) , Z = ( Z 1 , Z 2 ) ∈ Ω$ be such that the disks $B ⁢ ( Y ) , B ⁢ ( Z )$ are compactly supported in Ω. Let $( u 𝐘 , p 𝐘 ) , ( u 𝐙 , p 𝐙 ) ∈ ( H 0 1 ⁢ ( Ω ) ) 2 × L 0 2 ⁢ ( Ω )$ be the weak solutions of

${ - div ⁡ ( 2 ⁢ μ 𝒀 ⁢ D ⁢ ( 𝐮 𝒀 ) ) + ∇ ⁡ p 𝒀 = ρ 𝒀 ⁢ 𝐠 , div ⁡ 𝐮 𝒀 = 0 ,$

${ - div ⁡ ( 2 ⁢ μ 𝒁 ⁢ D ⁢ ( 𝐮 𝒁 ) ) + ∇ ⁡ p 𝒁 = ρ 𝒁 ⁢ 𝐠 , div ⁡ 𝐮 𝒁 = 0 ,$

where $μ 𝐘 , μ 𝐙 , ρ 𝐘 , ρ 𝐙$ are defined as in (2.2). If $| 𝐘 - 𝐙 | d ⁢ ( B ⁢ ( 𝐘 ) ; ∂ ⁡ Ω )$ is small enough, then there exists a constant $C > 0$ depending only on Ω such that

(B.3)

$| 1 π ⁢ R 2 ⁢ ∫ ℬ ⁢ ( 𝒀 ) 𝐮 𝒀 ⁢ 𝑑 𝐱 - 1 π ⁢ R 2 ⁢ ∫ ℬ ⁢ ( 𝒁 ) 𝐮 𝒁 ⁢ 𝑑 𝐱 | ≤ C ⁢ L ⁢ | 𝒀 - 𝒁 | d ⁢ ( ℬ ⁢ ( 𝒀 ) ; ∂ ⁡ Ω ) ,$

where

$L = ρ ℬ ⁢ | g | ε ⁢ R ⁢ μ ℱ 2 .$

The key point to prove this lemma consists in mapping $ℬ ⁢ ( 𝒀 )$ into $ℬ ⁢ ( 𝒁 )$ as in [11]. For details we refer to [6, Appendix B]. The fact that estimate (B.3) blows up when $d ⁢ ( ℬ ⁢ ( 𝑿 ) ; ∂ ⁡ Ω )$ approaches zero seems reasonable; it is due to the mapping from $ℬ ⁢ ( 𝒀 )$ into $ℬ ⁢ ( 𝒁 )$ which becomes singular. However, we do not know if the power one (on d) is sharp.

Our main result is then the following.

Theorem 2.

Let $( 𝐗 ⁢ ( t ) , u 𝐗 ⁢ ( t ) )$ , $t ∈ [ 0 , T ]$ , be the solution of (B.1) and $( 𝐗 n , u 𝐗 n )$ , $n = 0 , 1 ⁢ … , N$ , the solution of (B.2). Let τ be the maximal time step, assume that for all $n = 1 , 2 , 3 , … , N$ , $𝐗 n$ converges to $𝐗 ⁢ ( t n )$ as τ goes to 0. Then, if τ is small enough, there exists a constant $C > 0$ depending only on Ω such that

$| 𝑿 ⁢ ( t N ) - 𝑿 N | ≤ C ⁢ L ⁢ exp ⁡ ( ∫ 0 T C ⁢ L d ⁢ ( ℬ ⁢ ( 𝑿 ⁢ ( s ) ) ; ∂ ⁡ Ω ) ⁢ 𝑑 s ) ⁢ ∑ n = 0 N - 1 ∫ t n t n + 1 1 d ⁢ ( ℬ ⁢ ( 𝑿 ⁢ ( t ) ; ∂ ⁡ Ω ) ) ⁢ ( t - t n ) ⁢ | 𝑿 n + 1 - 𝑿 n τ n + 1 | ⁢ 𝑑 t .$

Proof.

Let $𝑿 τ$ be the piecewise linear reconstruction defined by

$𝑿 τ ⁢ ( t ) = 𝑿 n + ( t - t n ) ⁢ 𝑿 n + 1 - 𝑿 n τ n + 1 , t ∈ [ t n , t n + 1 ] , n = 0 , 1 , 2 , … , N - 1 .$

Let $e ⁢ ( t ) = 𝑿 ⁢ ( t ) - 𝑿 τ ⁢ ( t )$ , $n ≥ 0$ and $t ∈ ( t n , t n + 1 )$ . Using (B.1) and (B.2), we have

$𝒆 ˙ ⁢ ( t ) = 1 π ⁢ R 2 ⁢ ∫ ℬ ⁢ ( 𝑿 ⁢ ( t ) ) 𝐮 𝑿 ⁢ ( t ) ⁢ 𝑑 x - 1 π ⁢ R 2 ⁢ ∫ ℬ ⁢ ( 𝑿 n ) 𝐮 𝑿 n ⁢ 𝑑 x .$

Taking the scalar product of the last equation with $𝒆 ⁢ ( t )$ , using the Cauchy–Schwarz inequality and Lemma 1 yields

$1 2 ⁢ d d ⁢ t ⁢ | 𝒆 ⁢ ( t ) | 2 ≤ C ⁢ L d ⁢ ( ℬ ⁢ ( 𝑿 ⁢ ( t ) ) ; ∂ ⁡ Ω ) ⁢ | 𝑿 ⁢ ( t ) - 𝑿 n | ⁢ | 𝒆 ⁢ ( t ) | .$

Without loss of generality, we assume that $𝒆 ⁢ ( t ) ≠ 0$ , $t ∈ ( t n , t n + 1 )$ . Therefore, dividing by $| 𝒆 ⁢ ( t ) |$ , we get

$d d ⁢ t ⁢ | 𝒆 ⁢ ( t ) | = d d ⁢ t ⁢ | 𝒆 ⁢ ( t ) | 2 2 ⁢ | 𝒆 ⁢ ( t ) | ≤ C ⁢ L d ⁢ ( ℬ ⁢ ( 𝑿 ⁢ ( t ) ) ; ∂ ⁡ Ω ) ⁢ | 𝑿 ⁢ ( t ) - 𝑿 n |$

with $| 𝒆 ⁢ ( t ) | = | 𝒆 ⁢ ( t ) | 2$ . Applying the triangle inequality, we obtain

$d d ⁢ t ⁢ | 𝒆 ⁢ ( t ) | ≤ C ⁢ L d ⁢ ( ℬ ⁢ ( 𝑿 ⁢ ( t ) ) ; ∂ ⁡ Ω ) ⁢ | 𝑿 τ - 𝑿 n | + C ⁢ L d ⁢ ( ℬ ⁢ ( 𝑿 ⁢ ( t ) ) ; ∂ ⁡ Ω ) ⁢ | 𝒆 ⁢ ( t ) | ,$

Multiplying the last inequality by

$exp ⁡ ( - ∫ 0 t C ⁢ L d ⁢ ( ℬ ⁢ ( 𝑿 ⁢ ( s ) ) ; ∂ ⁡ Ω ) ⁢ 𝑑 s ) ,$

and integrating from $t n$ to $t n + 1$

$| 𝒆 ⁢ ( t n + 1 ) | ⁢ exp ⁡ ( - ∫ 0 t n + 1 C ⁢ L d ⁢ ( ℬ ⁢ ( 𝑿 ⁢ ( t ) ) ; ∂ ⁡ Ω ) ⁢ 𝑑 t ) - | 𝒆 ⁢ ( t n ) | ⁢ exp ⁡ ( - ∫ 0 t n C ⁢ L d ⁢ ( ℬ ⁢ ( 𝑿 ⁢ ( t ) ) ; ∂ ⁡ Ω ) ⁢ 𝑑 t )$

$≤ C ⁢ L ⁢ ∫ t n t n + 1 ( | 𝑿 τ ⁢ ( t ) - 𝑿 n | d ⁢ ( ℬ ⁢ ( 𝑿 ⁢ ( t ) ; ∂ ⁡ Ω ) ) ) ⁢ exp ⁡ ( - ∫ 0 t C ⁢ L d ⁢ ( ℬ ⁢ ( 𝑿 ⁢ ( s ) ) ; ∂ ⁡ Ω ) ⁢ 𝑑 s ) ⁢ 𝑑 t .$

Summing up over n leads to

$| 𝒆 ⁢ ( T ) | ≤ C ⁢ L ⁢ ∑ n = 0 N - 1 ∫ t n t n + 1 | 𝑿 τ ⁢ ( t ) - 𝑿 n | d ⁢ ( ℬ ⁢ ( 𝑿 ⁢ ( t ) ; ∂ ⁡ Ω ) ) ⁢ exp ⁡ ( ∫ t T C ⁢ L d ⁢ ( ℬ ⁢ ( 𝑿 ⁢ ( s ) ) ; ∂ ⁡ Ω ) ⁢ 𝑑 s ) ⁢ 𝑑 t ,$

where we use that $e ⁢ ( 0 ) = 0$ to get rid of the error at $t = 0$ . The desired estimate is then obtained by using the definition of $𝑿 τ$ . ∎

Acknowledgements

Frédéric Alauzet is acknowledged for providing the Wolf-Interpol program corresponding to conservative interpolation [3]. Samuel Dubuis acknowledges Diane Guignard and Swarnendu Sil for advices and discussions concerning the derivation of the time error indicator. It should be noted that some numerical experiments with space adaptation only were already reported in [26].

References

[1] M. Ainsworth, J. Z. Zhu, A. W. Craig and O. C. Zienkiewicz, Analysis of the Zienkiewicz–Zhu a posteriori error estimator in the finite element method, Internat. J. Numer. Methods Engrg. 28 (1989), no. 9, 2161–2174. 10.1002/nme.1620280912Search in Google Scholar

[2] F. Alauzet and A. Loseille, A decade of progress on anisotropic mesh adaptation for computational fluid dynamics, Comput. Aided Design 72 (2016), 13–39. 10.1016/j.cad.2015.09.005Search in Google Scholar

[3] F. Alauzet and M. Mehrenberger, $𝐏 1$ -conservative solution interpolation on unstructured triangular meshes, Internat. J. Numer. Methods Engrg. 84 (2010), no. 13, 1552–1588. 10.1002/nme.2951Search in Google Scholar

[4] Y. C. Chang, T. Y. Hou, B. Merriman and S. Osher, A level set formulation of Eulerian interface capturing methods for incompressible fluid flows, J. Comput. Phys. 124 (1996), no. 2, 449–464. 10.1006/jcph.1996.0072Search in Google Scholar

[5] T. Coupez, G. Jannoun, N. Nassif, H. C. Nguyen, H. Digonnet and E. Hachem, Adaptive time-step with anisotropic meshing for incompressible flows, J. Comput. Phys. 241 (2013), 195–211. 10.1016/j.jcp.2012.12.010Search in Google Scholar

[6] S. Dubuis, Adaptive algorithms for two fluids flows with anisotropic finite elements and order two time discretizations, PhD thesis, Ecole Polytechnique fédérale de Lausanne, 2019. Search in Google Scholar

[7] S. Dubuis and M. Picasso, An adaptive algorithm for the time dependent transport equation with anisotropic finite elements and the Crank–Nicolson scheme, J. Sci. Comput. 75 (2018), no. 1, 350–375. 10.1007/s10915-017-0537-1Search in Google Scholar

[8] L. P. Franca and S. L. Frey, Stabilized finite element methods. II. The incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg. 99 (1992), no. 2–3, 209–233. 10.1016/0045-7825(92)90041-HSearch in Google Scholar

[9] R. Glowinski, T. W. Pan, T. I. Hesla, D. D. Joseph and J. Périaux, A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow, J. Comput. Phys. 169 (2001), no. 2, 363–426. 10.1006/jcph.2000.6542Search in Google Scholar

[10] C. Grandmont and Y. Maday, Existence de solutions d’un problème de couplage fluide-structure bidimensionnel instationnarie, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 4, 525–530. 10.1016/S0764-4442(97)89804-7Search in Google Scholar

[11] D. Guignard, F. Nobile and M. Picasso, A posteriori error estimation for the steady Navier–Stokes equations in random domains, Comput. Methods Appl. Mech. Engrg. 313 (2017), 483–511. 10.1016/j.cma.2016.10.008Search in Google Scholar

[12] M. Hillairet, Lack of collision between solid bodies in a 2D incompressible viscous flow, Comm. Partial Differential Equations 32 (2007), no. 7–9, 1345–1371. 10.1080/03605300601088740Search in Google Scholar

[13] K.-H. Hoffmann and V. N. Starovoitov, On a motion of a solid body in a viscous fluid. Two-dimensional case, Adv. Math. Sci. Appl. 9 (1999), no. 2, 633–648. Search in Google Scholar

[14] H. H. Hu, N. A. Patankar and M. Y. Zhu, Direct numerical simulations of fluid-solid systems using the arbitrary Lagrangian–Eulerian technique, J. Comput. Phys. 169 (2001), no. 2, 427–462. 10.1006/jcph.2000.6592Search in Google Scholar

[15] J. A. Janela, A. Lefebvre and B. Maury, A penalty method for the simulation of fluid-rigid body interaction, CEMRACS 2004—Mathematics and Applications to Biology and Medicine, ESAIM Proc. 14, EDP Science, Les Ulis (2005), 115–123. 10.1051/proc:2005010Search in Google Scholar

[16] V. John and J. Rang, Adaptive time step control for the incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg. 199 (2010), no. 9–12, 514–524. 10.1016/j.cma.2009.10.005Search in Google Scholar

[17] P. Laug and H. Borouchaki, The BL2D mesh generator: Beginner’s guide, user’s and programmer’s manual, Technical report RT-0194, Institut National de Recherche en Informatique et Automatique (INRIA), Le Chesnay-Rocquencourt, 1996. Search in Google Scholar

[18] A. Lozinski, M. Picasso and V. Prachittham, An anisotropic error estimator for the Crank–Nicolson method: Application to a parabolic problem, SIAM J. Sci. Comput. 31 (2009), no. 4, 2757–2783. 10.1137/080715135Search in Google Scholar

[19] B. Maury, Numerical analysis of a finite element/volume penalty method, SIAM J. Numer. Anal. 47 (2009), no. 2, 1126–1148. 10.1007/978-1-4020-8758-5_9Search in Google Scholar

[20] S. Micheletti and S. Perotto, Space-time adaptation for purely diffusive problems in an anisotropic framework, Int. J. Numer. Anal. Model. 7 (2010), no. 1, 125–155. Search in Google Scholar

[21] S. Micheletti, S. Perotto and M. Picasso, Stabilized finite elements on anisotropic meshes: A priori error estimates for the advection-diffusion and the Stokes problems, SIAM J. Numer. Anal. 41 (2003), no. 3, 1131–1162. 10.1137/S0036142902403759Search in Google Scholar

[22] M. Picasso, Adaptive finite elements for a linear parabolic problem, Comput. Methods Appl. Mech. Engrg. 167 (1998), no. 3–4, 223–237. 10.1016/S0045-7825(98)00121-2Search in Google Scholar

[23] M. Picasso, An anisotropic error indicator based on Zienkiewicz–Zhu error estimator: Application to elliptic and parabolic problems, SIAM J. Sci. Comput. 24 (2003), no. 4, 1328–1355. 10.1137/S1064827501398578Search in Google Scholar

[24] M. Picasso, An adaptive algorithm for the Stokes problem using continuous, piecewise linear stabilized finite elements and meshes with high aspect ratio, Appl. Numer. Math. 54 (2005), no. 3–4, 470–490. 10.1016/j.apnum.2004.09.014Search in Google Scholar

[25] J. A. San Martín, V. Starovoitov and M. Tucsnak, Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Ration. Mech. Anal. 161 (2002), no. 2, 113–147. 10.1007/s002050100172Search in Google Scholar

[26] N. Sauerwald, An adaptive method for solving stokes’ flow around a falling sphere, Master’s thesis, Institute of Mathematics, 2014. Search in Google Scholar

[27] V. N. Starovoitov, Behavior of a rigid body in an incompressible viscous fluid near a boundary, Free Boundary Problems (Trento 2002), Internat. Ser. Numer. Math. 147, Birkhäuser, Basel (2004), 313–327. 10.1007/978-3-0348-7893-7_25Search in Google Scholar

[28] T. Takahashi, Existence of strong solutions for the problem of a rigid-fluid system, C. R. Math. Acad. Sci. Paris 336 (2003), no. 5, 453–458. 10.1016/S1631-073X(03)00081-5Search in Google Scholar

[29] N. Verdon, A. Lefebvre-Lepot, L. Lobry and P. Laure, Contact problems for particles in a shear flow, Eur. J. Comput. Mech. 19 (2010), no. 5–7, 513–531. 10.3166/ejcm.19.513-531Search in Google Scholar

[30] O. C. Zienkiewicz and J. Z. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis, Internat. J. Numer. Methods Engrg. 24 (1987), no. 2, 337–357. 10.1002/nme.1620240206Search in Google Scholar

[31] O. C. Zienkiewicz and J. Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. I. The recovery technique, Internat. J. Numer. Methods Engrg. 33 (1992), no. 7, 1331–1364. 10.1002/nme.1620330702Search in Google Scholar

Received: 2020-03-31

Revised: 2020-07-31

Accepted: 2020-10-23

Published Online: 2020-11-07

Published in Print: 2021-04-01

You are currently not able to access this content.

Articles in the same Issue

DOI: doi.org/10.1515/cmam-2020-0046

Keywords for this article

Anisotropic Finite Elements; Adaptive Method; Solid-Fluid Interactions