Boundary and interface methods for energy stable finite difference discretizations of the dynamic beam equation

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Boundary and interface methods for energy stable

ﬁnite diﬀerence discretizations of the dynamic beam

equation

Gustav Eriksson∗, Jonatan Werpers, David Niemelä, Niklas Wik, Valter

Zethrin, Ken Mattsson

Department of Information Technology, Uppsala University, PO Box 337, S-751 05

Uppsala, Sweden

Abstract

We consider energy stable summation by parts ﬁnite diﬀerence methods

(SBP-FD) for the homogeneous and piecewise homogeneous dynamic beam

equation (DBE). Previously the constant coeﬃcient problem has been solved

with SBP-FD together with penalty terms (SBP-SAT) to impose boundary

conditions. In this work we revisit this problem and compare SBP-SAT to the

projection method (SBP-P). We also consider the DBE with discontinuous

coeﬃcients and present novel SBP-SAT, SBP-P and hybrid SBP-SAT-P dis-

cretizations for imposing interface conditions. Numerical experiments show

that all methods considered are similar in terms of accuracy, but that SBP-P

can be more computationally eﬃcient (less restrictive time step requirement

for explicit time integration methods) for both the constant and piecewise

constant coeﬃcient problems.

Keywords: dynamic beam equation, boundary treatment, summation by

parts, ﬁnite diﬀerences, high order methods

1. Introduction

The dynamic beam equation (DBE) is a standard beam theory model

describing the motion of free vibrations of a Euler-Bernoulli beam. The

∗Corresponding author

E-mail address: gustav.eriksson@it.uu.se

Preprint submitted to Journal of Computational Physics October 25, 2022

arXiv:2210.13131v1 [math.NA] 24 Oct 2022

derivation of the equation dates back to the 18th century but it is still used in

engineering applications today. For example, in construction of infrastructure

involving beams such as buildings, bridges and railways.

Most of the literature on the DBE focus on solving the constant coef-

ﬁcient problem in the frequency domain. By taking the Laplace transform

of (1) one can, for speciﬁc boundary conditions and external loads, derive

the fundamental frequencies of the beam and thus obtain explicit analytical

solutions [1, 2, 3, 4]. Although this approach is highly eﬃcient when appli-

cable, the analysis becomes signiﬁcantly more complex when discontinuous

beam parameters and general dynamic loads are required. An alternative is

to utilize numerical methods. In [5] the spatial component of the solution is

discretized with compact ﬁnite diﬀerences and used to numerically derive the

natural frequencies of nanobeams. However, this approach assumes that the

spatial and temporal parts of the solution function can be separated. In this

paper we solve the DBE numerically in the time-domain using the method

of lines, i.e. we discretize the partial diﬀerential equation (PDE) in space

and then solve the resulting system of ordinary diﬀerential equations (ODE).

The spatial discretization is done using high order explicit ﬁnite diﬀerence

methods. This approach is not limited by any speciﬁc choice of boundary

condition or external loads, and does not assume that the solution takes any

particular form (e.g. separate temporal and spatial components).

It has for a long time been known that non-trivial boundary procedures

are required to obtain stable high order ﬁnite diﬀerence discretizations of

initial-boundary value problems (IBVP) [6, 7, 8, 9, 10]. One way to manage

this is to combine summation by parts (SBP) ﬁnite diﬀerence operators with

either the simultaneous approximation term (SAT) method or the projection

(P) method to impose boundary conditions. With both SBP-SAT or SBP-P

a semi-discrete energy estimate that mimics the continuous estimate can be

derived, ensuring stability of the ODE system. The SBP-SAT and SBP-P

methods have been applied to various PDE:s in the past, see for example

[11, 12, 13, 14, 15]. See also [16, 17, 18] for examples of the SBP-SAT

method applied to IBVP:s involving third and fourth derivatives. In [16] the

SBP-SAT method was used to solve the DBE with constant coeﬃcients.

In this paper we begin by deriving stable SBP-P discretizations of the

DBE with clamped and free boundary conditions, and compare them to pre-

viously presented SBP-SAT discretizations [16]. We then consider the DBE

with piecewise constant coeﬃcients, requiring additional interface conditions

to couple the solution across the discontinuities. We present novel SBP-SAT,

SBP-P and hybrid SBP-SAT-P discretizations for this problem and compare

them numerically in terms of accuracy and computational eﬃciency. Previ-

ously SBP-SAT and SBP-P have been used to impose interface conditions

for PDE:s with second derivatives in space, see for example [12, 14, 19]. Here

we propose a novel third alternative utilizing SAT and projection simultane-

ously, SBP-SAT-P, where some interface conditions are imposed using SAT

and others using projection. The focus of the current work is twofold: a):

to develop a stable method for solving the DBE with general parameters,

boundary conditions and external loads and b): to compare SBP-SAT, SBP-

P and SBP-SAT-P for a problem involving high spatial derivatives (for which

it is traditionally diﬃcult to derive energy stable schemes).

This paper is organized as follows: In Section 2 the DBE is introduced.

In Section 3 necessary deﬁnitions are presented. In Sections 4 and 5 the DBE

with constant and piecewise constant coeﬃcients respectively is analyzed. In

Section 6 the time stepping scheme is presented. In Section 7 the methods

are compared and the analysis is veriﬁed numerically. Conclusions are drawn

in Section 8.

2. The dynamic beam equation

Denote by w(x, t)the beam’s deﬂection from the x-axis, the DBE is then

given by

µ(x)∂2w(x, t)

∂t2=−∂2

∂x2E(x)I(x)∂2w(x, t)

∂x2+q(x, t), xl≤x≤xr, t > 0,

w(x, t) = f1(x),∂w(x, t)

∂t =f2(x), xl≤x≤xr, t = 0,

BL(w) = GL(t), x =xl, t > 0,

BR(w) = GR(t), x =xr, t > 0,

(1)

where E(x)is the elastic modulus of the beam, I(x)the second moment of

area of the beam’s cross-section, µ(x)the mass per unit length and q(x, t)the

external load. The initial data is f1,2(x). The boundary operators BL,R and

boundary data GL,R(t)determine the boundary conditions. For notational

clarity, we rewrite the PDE as

b(x)utt =−(a(x)uxx)xx +F(x, t),(2)

where subscripts denote partial diﬀerentiation. Note that the functions

a(x) = E(x)I(x)and b(x) = µ(x)are positive. Throughout this paper

we will assume that the boundary data GL,R(t)and forcing function F(x, t)

are zero (all methods considered are equally applicable with general non-

homogeneous data).

3. Deﬁnitions

Let the inner product of two real-valued functions u, v ∈L2[xl, xr]be

deﬁned by (u, v) = Rxr

xluv dx and the corresponding norm by kuk2= (u, u).

The domain is discretized into mequidistant grid points given by

xi=xl+ (i−1)h, i = 1,2, ..., m, h =xr−xl

m−1.(3)

A semi-discrete solution vector is given by v= [v1, v2, ..., vm]>, where vi

is the approximate solution at grid point xi. Let (u, v)H=u>Hv, where

H=H>>0, deﬁne a discrete inner product for discrete real-valued vectors

u, v ∈Rm, and kvk2

H=v>Hv the corresponding norm. We also deﬁne

H=H0

0H.(4)

The SBP operators used here are central ﬁnite diﬀerence operators with

boundary closures carefully designed to mimic integration-by-parts in the

semi-discrete setting. In this paper we present results for the 2nd, 4th and

6th order accurate fourth-derivative SBP operators presented in [18], the

following deﬁnition is central:

Deﬁnition 1. A diﬀerence operator

D4=H−1N+eld>

3;l+d1;ld>

2;l+erd>

3;r−d1;rd>

2;rapproximating ∂4/∂x4, us-

ing a 2pth-order accurate narrow-stencil in the interior, is said to be a 2pth-

order diagonal-norm fourth-derivative SBP operator if H=H>>0is diag-

onal, N=NT≥0,d>

1;lv≈ −ux|l,d>

1;rv≈ux|r,d>

2;lv≈ −uxx|l,d>

2;rv≈uxx|r,

3;lv≈ −uxxx|land d>

3;rv≈uxxx|rare ﬁnite diﬀerence approximations of

the ﬁrst, second and third normal derivatives at the left and right boundary

points.

The matrix Ncan be decomposed in the following way:

N=˜

N+hαII d2;ld>

2;l+d2;rd>

2;r+h3αIII d3;ld>

3;l+d3;rd>

3;r,(5)

where ˜

N=˜

N>≥0and αII and αIII are positive constants not dependent

on h. In Figure 1 the values ensuring positive semi-deﬁniteness of ˜

Nare

presented for the 2nd, 4th and 6th order accurate SBP operators used here.

2nd order

0 0.5 1 1.5

0.2

0.4

0.6

0.8

1.2

4th order

0 0.5 1 1.5

0.2

0.4

0.6

0.8

1.2

0 0

6th order

0 0.5 1 1.5

0.2

0.4

0.6

0.8

1.2

Figure 1: Inﬂuence of parameters αII and αIII on deﬁniteness of ˜

4. Homogeneous beam

4.1. Well-posedness

We begin by considering a homogeneous beam, i.e. a(x)and b(x)are

constant functions. Multiplying (2) by ut, integrating over the domain and

using integration by parts (the energy method) leads to

dtE= 2a[−utuxxx +utxuxx]xr

xl,(6)

where Eis an energy deﬁned as

E=bkutk2+akuxxk2.(7)

Assuming existence, well-posedness of (2) is achieved by imposing the mini-

mum number of boundary conditions such that the right-hand side in (6) is

non-positive with zero boundary data, i.e. the energy for the homogeneous

problem is non-increasing. For this problem the minimum number of bound-

ary conditions are two on each boundary [16]. In this paper we consider the

following homogeneous boundary conditions:

clamped: u= 0, ux= 0, x =xl,r,

free: uxx = 0, uxxx = 0, x =xl,r.(8)

Either combination of these lead to energy conservation,

dtE= 0.(9)

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BoundaryandinterfacemethodsforenergystablenitedierencediscretizationsofthedynamicbeamequationGustavEriksson,JonatanWerpers,DavidNiemelä,NiklasWik,ValterZethrin,KenMattssonDepartmentofInformationTechnology,UppsalaUniversity,POBox337,S-75105Uppsala,SwedenAbstractWeconsiderenergystablesummationbypar...

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Boundary and interface methods for energy stable finite difference discretizations of the dynamic beam equation

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