Abstract
The stress-strain constitutive law for viscoelastic materials such as soft tissues, metals at high temperature, and polymers can be written as a Volterra integral equation of the second kind with a fading memory kernel. This integral relationship yields current stress for a given strain history and can be used in the momentum balance law to derive a mathematical model for the resulting deformation. We consider such a dynamic linear viscoelastic model problem resulting from using a Dirichlet–Prony series of decaying exponentials to provide the fading memory in the Volterra kernel. We introduce two types of internal variable to replace the Volterra integral with a system of auxiliary ordinary differential equations and then use a spatially discontinuous symmetric interior penalty Galerkin (SIPG) finite element method and – in time – a Crank–Nicolson method to formulate the fully discrete problems: one for each type of internal variable. We present a priori stability and error analyses without using Grönwall’s inequality and with the result that the constants in our estimates grow linearly with time rather than exponentially. In this sense, the schemes are therefore suited to simulating long time viscoelastic response, and this (to our knowledge) is the first time that such high quality estimates have been presented for SIPG finite element approximation of dynamic viscoelasticity problems. We also carry out a number of numerical experiments using the FEniCS environment (https://fenicsproject.org), describe a simulation using “real” material data, and explain how the codes can be obtained and all of the results reproduced.
1 Introduction
The application of a nonsymmetric interior penalty discontinuous Galerkin (NIPG) finite element method (DGFEM) to a dynamic linear solid viscoelasticity problem with tensor-valued internal variable stress representation was presented by Rivière, Shaw, and Whiteman in [21].
They gave an a priori energy error estimate by using the standard Grönwall inequality to deal with the time accumulation of error, and hence the constants in the stability and error bounds are too large to give confidence in the long time simulation of viscoelastic response.
In this paper, we use the symmetric interior penalty Galerkin (SIPG) method and prove stability bounds and a priori error estimates (not only in the energy norm but also in the spatial
We consider a linear homogeneous and isotropic viscoelastic solid material (see e.g. [11]) occupying a bounded polytopic domain, the interior of which is denoted by
where overdots denote time differentiation so that
for given functions
To close the problem and solve for displacement, we need a constitutive equation expressing stress in terms of displacement.
In the linear viscoelasticity model considered here, this involves a Volterra (or “fading memory”) integral with the specific material characterised by stiffness
where
Note that, in (1.5), we use the shorthand
with
This paper is arranged as follows. In Section 2, we give our notations and the preliminary background for DGFEM. In Section 3, we introduce two forms of internal variables, each of which are used to represent the Volterra (or “history”) integral, and formulate a variational problem for each form. We then state and prove (without using Grönwall’s inequality) stability bounds in Section 4 and error bounds in Section 5, carry out some illustrative numerical experiments in Section 6, using FEniCS (see [1] and https://fenicsproject.org), and then end with some concluding remarks in Section 7.
2 Preliminary
We use standard notation so that
for
When convenient, we shall often replace the upper limit 𝑇 in these expressions by some other value
For inner products of vector-valued and tensor-valued functions, we use the same notation as for the scalar cases. For instance, we have
for vector-valued functions 𝒗 and 𝒘, and second-order tensors
Meshes
We refer to [19] for a detailed explanation of the framework of the DGFEM and here just summarise the main points.
Assume that the closure of Ω is subdivided into closed elements 𝐸, where 𝐸 is a triangle in 2D or a tetrahedron in 3D, and the intersection of any pair of elements is either a vertex, an edge, a face, or empty.
The diameter of 𝐸 is defined by
Next, let
Test Spaces
We introduce the broken Sobolev space
We note the following facts:
We define the space of polynomials of degree less than or equal to 𝑘 on 𝐸, for
and then define our DG finite element space as
The analogous vector field is given by
Average and Jump
Suppose two elements
where the normal vector
We can now introduce the jump penalty operator
where
Useful Inequalities
We now recall the following inequalities for use later in the a priori analysis.
Inverse polynomial trace inequalities [23]: For any
(2.1)where 𝐶 is a positive constant and is independent of
Inverse inequality (or Markov inequality) [17, 19]: For any
(2.3)where
DG Bilinear Form
We define a DG bilinear form
for any
Comparing these, we can observe that
In the DG bilinear form, the third term is called the “interior penalty” term and the last one is called the “jump penalty”.
Depending on the sign of the interior penalty, the bilinear form is either symmetric or nonsymmetric.
In this article, we consider only the symmetric DG method and refer to [15, 21] for an application of the nonsymmetric method for viscoelasticity.
The reason why we employ SIPG is that it requires only the standard penalisation
Remark 2.2 (Korn’s Inequality for Piecewise
H
1
Vector Fields [4, 19])
If we have
for some positive 𝐶 independent of 𝒗.
In CGFEM, the squared energy norm
Suppose
where 𝑒 is the shared edge of elements
Proof
Let
after noting that
for a positive constant 𝐶.
Since
because
for any positive 𝜖.
Taking
By (2.8), we can also derive
where 𝐶 is a positive constant independent of
Theorem 2.1 (Coercivity and Continuity)
Suppose
where 𝜅 and 𝐾 are independent of 𝒗 and 𝒘.
Proof
The proof follows the same arguments as in [15, Theorems 4.5 and 4.6]. For example, the use of (2.8) and (2.9) leads us to show the coercivity and the continuity. For details, please see [15]. ∎
Due to the use of inverse polynomial trace inequality, the DG bilinear form will not be coercive and continuous on the broken Sobolev space.
In other words, Theorem 2.1 holds only on the finite element space.
For the choice of the penalty parameter
DG Elliptic Projection
The DG elliptic projector 𝑹 is defined for
Referring to [19, 20, 13, 24], for example, we recall the following elliptic-error estimates:
We now move on to describe the model problem.
3 Model Problem
We recall our primal equations (1.1)–(1.6) and hereafter assume the data terms are bounded and smooth so that
3.1 Internal Variables
Recalling the constitutive equation (1.5) and the stress relaxation function (1.6), we define internal variables as
for
Displacement Form
We consider (1.1) in the following form: find
Velocity Form
We consider (1.1) in the form
The (well-known) idea here is to replace the integral form in the constitutive hereditary laws by ODEs for a set of internal variables. With these, we use a time stepper rather than numerical integration to compute the discrete solution. Furthermore, while [21] employed internal variables as second-order tensors, we have defined vector-valued internal variables. This reduces the computer memory requirement by a factor of 𝑑.
3.2 Variational Formulation
We begin by multiplying each of (3.4) and (3.6) by test functions
Weak Forms
Let us define linear forms by
Recalling (2.4), we now use these linear forms to formulate variational problems for the displacement form and the velocity form. First for the displacement form of the problem.
Displacement Form Problem (D)
Find 𝒖 and
Velocity Form Problem (V)
Find 𝒖 and
Note that, in (3.9) and (3.11), the energy bilinear form, rather than the
Fully Discrete Formulations
For the time discretisation, we employ a Crank–Nicolson type finite difference scheme [7].
Let
and we will impose the relation
We can now give the fully discrete schemes.
Displacement Form Problem (D)ℎ
Find
and secondly that, for
with initial data
Velocity Form Problem (V)ℎ
Find
and secondly that, for
with initial data (3.15), (3.16),
and where
The jump term penalisation of the velocities in (3.8) and (3.10) is included in these formulations to facilitate the error analysis later: see (5.10) in particular. This term is not required for the stability estimates that follow.
4 Stability Analysis
In this section, we derive stability bounds for the solutions to the fully discrete displacement and velocity forms of the problem.
As the problems are linear, these stability bounds can also be used to conclude the existence and uniqueness of the numerical solutions and, hence, the well-posedness of the discrete problems.
At first, we present bounds that are non-optimal in that they contain a factor of
The
Theorem 4.1 (Stability Bound for (D)ℎ)
If
Here, 𝐶 is independent of the discrete solutions,
Proof
Choose
since, by (3.12),
which, when used with (3.14), gives us
for each 𝑛 and 𝑞. The cancellation of terms in this step illustrates the importance of enforcing the internal variable ODEs in the energy bilinear form in (3.9) and (3.11). Using (2.6), (4.1), and Theorem 2.1 in (4.1), we obtain
Now we show an upper bound for the right-hand side of (4.2).
hence
by (3.15). This implies
for any positive
By (3.12) and the Cauchy–Schwarz inequality, we therefore have
Since the inverse polynomial trace inequality (2.1) and Poincaré’s inequality (2.2) imply that
(4.3)the triangle inequality and Young’s inequality lead us to
for any positive
When we set
inequality (4.4) gives
After noting that the right-hand side of (4.5) is independent of 𝑚 and that 𝑚 is arbitrary, we can obtain
where
Theorem 4.2 (Stability Bound for (V)ℎ)
If
Here, 𝐶 is independent of discrete solutions,
Proof
The proof follows the same arguments as in Theorem 4.1.
We take
which can be dealt with by using the fundamental theorem of calculus, summation by parts, and continuity. Specifically,
since
As we have already touched upon above, the
For a constant
where
Corollary 4.1 (to Theorems 4.1 and 4.2)
With the techniques and result from [5] just discussed, in Theorems 4.1 and 4.2, the terms
Proof
By
for all
Again, we emphasise that obtaining these improved results requires lengthy and non-trivial arguments which it would not be appropriate to try and replicate here. The full details are of course in [5].
5 Error Analysis
In order to carry out an a priori error analysis, we take the usual approach of splitting the error into “spatial” and “temporal” components by using the elliptic projection.
To this end, define
for
Also, for a three-times time-differentiable function
Hence, if
As is well known, the usual path to an error bound is to use the triangle inequality to split the error using the spatial and temporal components introduced above. The spatial errors are bounded using standard results for elliptic problems, and so it is the temporal error components that demand the bulk of the effort. For our problems, this effort is contained in the next two lemmas: the first for (D)ℎ and the second for (V)ℎ.
Suppose
where
Proof
The proof will follow similar arguments in the stability analysis, but we additionally use Galerkin orthogonality, the elliptic error estimates (2.10) and (2.11), and the time discretisation error (5.2).
Averaging over
for all
by Galerkin orthogonality, for any
Inserting
Using the elliptic projection, (3.15), (3.16), and initial conditions, we have
by the Cauchy–Schwarz inequality, Young’s inequality, and (5.2).
On the other hand, the elliptic error estimates such as (2.11) provide spatial error estimates for
Here, 𝐶 is independent of ℎ,
Next, the analogous estimate for (V)ℎ.
Suppose
where
Proof
We follow similar steps to the proof of Lemma 5.1 for the displacement form to show this claim.
By averaging at
where
We complete the proof by using the same techniques and results as used earlier: the Cauchy–Schwarz inequality and Young’s inequality, integration and summation by parts, (2.9), (2.11), and (5.2). ∎
We can now use Lemmas 5.1 and 5.2 to prove the following a priori error estimates.
Theorem 5.1 (Error Analysis)
Suppose the discrete solutions in
where
Proof
Using the triangle inequality with Lemmas 5.1 and 5.2, and the DG elliptic error estimates, we get for any
Poincaré’s inequality (2.2) indicates that
In a similar way, we have
On the other hand, as seen in (5.5) and (5.6),
the jump penalty term of
Using this argument with the inverse inequality (2.3), we can derive energy norm error estimates for the velocity vector field and get
Since 𝑛 is arbitrary and the right-hand sides of (5.7)–(5.11) are independent of 𝑛, the proof is completed.
Furthermore, to show the discrete errors for
by the fact
6 Numerical Experiments
In this section, we use the FEniCS environment (see [1] and https://fenicsproject.org) to give the results of some numerical experiments that were carried out to demonstrate the convergence rates proven above: we present tables of numerical errors, as well as convergence rates. The Python code to reproduce these results is available at Jang’s GitHub (https://github.com/Yongseok7717/Visco_Int_CG) and can be used to reproduce the results that are given below. Alternatively, these results can also be reproduced by using Docker. In this case, the commands to run are docker pull jangyour/fenics_fem docker run -ti jangyour/fenics_fem cd visco/Int_DG; . main.sh
An Exact Solution
As is common, we use an artificial, or manufactured, exact solution in order to calculate the errors. Let this manufactured strong solution be
where
and then
Numerical Results
We present numerical errors to demonstrate the convergence rates given by the theorems, and for this, we define
since
for some positive constant 𝐶.
Since the DG energy norm depends on the penalty parameters
Firstly, we set
We can see in Figure 1 that
Numerical convergence order: linear (top) and quadratic (bottom) polynomial basis
Secondly, if we take
Thirdly, in Table 3, we give some results for when the spatial error became negligibly small to the temporal error, to observe second-order accuracy in time regardless of spatial norms, displacement/velocity fields, and internal variable forms. All of these results are consistent with the claims in the theorems.
| 𝑘 | ℎ | Displacement form | Velocity form | ||
|---|---|---|---|---|---|
|
|
|||||
|
|
|
|
|
||
| 1 | 1/4 | 1.298e−01 | 1.951e−01 | 1.298e−01 | 1.951e−01 |
| 1/8 | 6.177e−02 (1.07) | 8.741e−02 (1.16) | 6.177e−02 (1.07) | 8.741e−02 (1.16) | |
| 1/16 | 2.993e−02 (1.05) | 4.130e−02 (1.08) | 2.993e−02 (1.05) | 4.130e−02 (1.08) | |
| 1/32 | 1.473e−02 (1.02) | 2.001e−02 (1.04) | 1.473e−02 (1.02) | 2.001e−02 (1.04) | |
|
|
|||||
| 2 | 1/4 | 3.168e−03 | 4.996e−03 | 3.168e−03 | 4.996e−03 |
| 1/8 | 8.030e−04 (1.98) | 1.284e−03 (1.96) | 8.030e−04 (1.98) | 1.284e−03 (1.96) | |
| 1/16 | 2.008e−04 (2.00) | 3.256e−04 (1.98) | 2.008e−04 (2.00) | 3.256e−04 (1.98) | |
| 1/32 | 5.010e−05 (2.00) | 8.206e−05 (1.99) | 5.010e−05 (2.00) | 8.206e−05 (1.99) | |
| 𝑘 | ℎ | Displacement form | Velocity form | ||
|---|---|---|---|---|---|
|
|
|||||
|
|
|
|
|
||
| 1 | 1/4 | 1.067e−02 | 2.293e−02 | 1.067e−02 | 2.293e−02 |
| 1/8 | 2.808e−03 (1.93) | 6.691e−03 (1.78) | 2.808e−03 (1.93) | 6.691e−03 (1.78) | |
| 1/16 | 7.094e−04 (1.98) | 1.182e−03 (1.88) | 7.094e−04 (1.98) | 1.182e−03 (1.88) | |
| 1/32 | 1.781e−04 (1.99) | 4.686e−04 (1.95) | 1.781e−04 (1.99) | 4.686e−04 (1.95) | |
|
|
|||||
| 2 | 1/4 | 8.362e−05 | 1.496e−04 | 8.362e−05 | 1.496e−04 |
| 1/8 | 1.011e−05 (3.05) | 1.861e−05 (3.01) | 1.011e−05 (3.05) | 1.861e−05 (3.01) | |
| 1/16 | 1.231e−06 (3.04) | 2.315e−06 (3.01) | 1.231e−06 (3.04) | 2.315e−06 (3.01) | |
| 1/32 | 1.515e−07 (3.02) | 2.906e−07 (3.00) | 1.514e−07 (3.02) | 2.902e−07 (3.00) | |
Time convergence errors (orders) for fixed
| Norm |
|
Displacement form | Velocity form | ||
|---|---|---|---|---|---|
|
|
|||||
|
|
|
|
|
||
|
|
1/2 | 2.696e−02 | 9.579e−02 | 1.766e−02 | 7.348e−02 |
| 1/4 | 7.445e−03 (1.86) | 2.461e−02 (1.96) | 4.879e−03 (1.86) | 1.880e−02 (1.97) | |
| 1/8 | 1.894e−03 (1.97) | 6.169e−03 (2.00) | 1.2429e−03 (1.97) | 4.712e−03 (2.00) | |
| 1/16 | 4.747e−04 (2.00) | 1.549e−04 (2.00) | 3.117e−04 (2.00) | 1.181e−03 (2.00) | |
|
|
|||||
|
|
1/2 | 8.121e−02 | 3.222e−02 | 5.256e−03 | 2.586e−02 |
| 1/4 | 2.410e−03 (1.75) | 8.221e−03 (1.97) | 1.534e−03 (1.78) | 6.601e−03 (1.97) | |
| 1/8 | 6.252e−04 (1.95) | 2.067e−03 (1.99) | 3.974e−04 (1.95) | 1.659e−03 (1.99) | |
| 1/16 | 1.577e−04 (1.99) | 5.178e−04 (2.00) | 1.001e−04 (1.99) | 4.155e−04 (2.00) | |
Penalty Parameters
For well-posedness as well as optimal convergence, we need to have “large enough” penalty parameters, and following studies of elasticity problems [13, 24], we took
| Displacement form | Velocity form | |||||
|---|---|---|---|---|---|---|
|
|
||||||
| ℎ | 1/2 | 1/4 | 1/8 | 1/2 | 1/4 | 1/8 |
| Error | 2.262 | 9.527e+03 | 1.262e+14 | 2.286 | 9.364e+03 | 1.266e+14 |
An Example with Real Material Data
The results above are based on artificial choices of the material data – the point was merely to show convergence and illustrate the estimates. In this closing section, we illustrate the displacement formulation using real material data for PMMA (polymethyl methacrylate), taken from [18, Table 1]. The demonstration given below shows clearly the effect of viscoelastic damping on the longer time response.
First, the Prony series relaxation data from [18, Table 1]
are converted from dyne/cm2 to N/m2 by dividing by ten, and then the coefficients are normalised (and nondimensionalised) to fit our assumptions for (1.6).
The results of this are in Table 5.
The stiffness modulus resulting from this normalisation turns out to be
for Lamé parameters
Prony series data for PMMA taken from [18, Table 1].
The second and third columns are from the source, one tenth of the sum of the third column (because
| 𝑞 |
|
dyne/cm2 |
|
|---|---|---|---|
| 0 |
|
0.001000236663139 | |
| 1 | 0.02 |
|
0.086627639575435 |
| 2 | 0.20 |
|
0.126369185566228 |
| 3 | 2 |
|
0.247379960437068 |
| 4 | 20 |
|
0.268813603218619 |
| 5 | 200 |
|
0.173255279150871 |
| 6 | 2000 |
|
0.069659339040041 |
| 7 | 20000 |
|
0.018307903209241 |
| 8 | 200000 |
|
0.006162172299696 |
| 9 | 2000000 |
|
0.001643245946586 |
| 10 | 20000000 |
|
0.000352762037446 |
| 11 | 200000000 |
|
0.000428672855631 |
We take
for
We give two different solutions to compare elastic and viscoelastic short time response to the traction.
The elastic solution is generated by setting
Snapshots at
Plots of the discrete solution
The code that produced these results can be obtained from https://github.com/variationalform/SIPG_PMMA_demo_2022 with git clone git@github.com:variationalform/SIPG_PMMA_demo_2022.git Or one can generate the results in a Docker container from https://hub.docker.com/r/variationalform/fem with docker pull variationalform/fem:SIPG_PMMA_demo_2022 docker run -ti variationalform/fem:SIPG_PMMA_demo_2022 # in the container cd runtime/le ./longrun_le.sh | tee ./longrun_le_out.txt for the linear elastic runs. Alter le to ve to run with the viscoelastic data. This Docker image uses FEniCS with DOLFIN version 2017.2.0.
7 Conclusion and Discussion
We have presented an a priori stability and error analysis of the DGFEM for linear dynamic viscoelasticity with two types of internal variables used to replace the fading memory Volterra integral. We have given illustrative numerical results, explained how these results can be reproduced, and we have observed that both of the schemes (for each type of internal variable) behave similarly. The numerics are consistent with the predicted convergence rates.
The main achievement of this work is that we have been able to give the stability and error bounds without recourse to Grönwall’s inequality, with the result that the stability and error bounds depend explicitly on the final time linearly, and not exponentially (along with the time dependence in the exact solution norms). This is a significant improvement over the estimates given in [21]. Moreover, the schemes presented here use displacement or velocity internal variables rather than stresses, and so – compared to [21] – for the schemes presented here, there is a modest reduction in the computer memory requirements.
An inspection of Tables 1, 2, and 3 indicates that there is no particular reason to prefer one formulation over the other. This is perhaps not surprising, but we are not aware of any previous work that has demonstrated it both in theoretical estimates and in numerical experiments. We return to this point below.
We assumed throughout that
where we have set
Finally, the results in Figures 2 and 3 demonstrate that this method can be applied to real materials.
Acknowledgements
Y. Jang acknowledges the support of a Brunel University London Doctoral scholarship.
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