A temporally piecewise adaptive multiscale scaled boundary finite element

method to solve two-dimensional heterogeneous viscoelastic problems

ARTICLE INFO

Keywords:

Reduced order

Heterogeneous viscoelastic materials

Multiscale SBFEM

Numerical base function

Temporally piecewise adaptive algorithm

ABSTRACT

By combining the Multiscale Scale Boundary Finite Element Method (MsSBFEM) and the Temporally Piecewise

Adaptive Algorithm (TPAA), a new numerical model is presented to reduce the solution scale of the twodimensional heterogeneous viscoelastic problems. Utilizing TPAA, a spatially and temporally related problem

is transformed into a series of recursive spatial problems, which are solved by MsSBFEM. The solution scale can

be effectively reduced through the use of a Scaled Boundary Finite bridge between the small and large scales. Element Method (SBFEM)-based numerical base functions, and the only way to improve the solution accuracy is by increasing the number of coarse element nodes without adding any new ones. Because of the singular, polygon, and SBFEM, or quadtree elements, makes the proposed algorithm more effective and makes it easier to work with the emphasize singularity and create a SBFE mesh. A safeguard for the temporal computational is provided by TPAA. accuracy through an adaptive procedure when the size of the steps changes. To make the point, numerical examples are provided. effectiveness of proposed strategies, and both large and small-scale outcomes are satisfactory.

1. Introduction

Direct Numerical Simulation (FE-DNS) for Heterogeneous Systems Based on FEM problems caused by viscoelasticity frequently involve a large number of required DOF. for discretization in accordance with the distribution of diverse materials. For instance, the degree of freedom (DOF) of pixel/ in the DNS-based effective evaluation on RVE for heterogeneous viscoelastic materials 107 will scale with FE-DNS based on voxels [1]. On the other hand, as a result of the materials heterogeneity, the generation of FE meshes is difficult because of with irregular geometry of constitutes and transition between the areas

with various mesh densities to meet the requirements of the interface between element and material [2]. Additionally, when cracks are present, the In the heterogeneous analysis, stress singularity becomes a problem. a method that is quick and easy to use. The temporal solution is required because viscoelastic analysis is time-dependent and typically involves step marching. accuracy must be maintained for various step sizes, which typically difficult to predict. As a result, effective numerical algorithms are unquestionably required after thorough consideration of the solution's reduction the temporal solution, scale, mesh generation, and stress singularity accuracy in addition. However, in the context of numerical methods based on FE-DNS, analysis of problems with heterogeneous viscoelastics, there appear to be few reports

directly related to this issue.

Hou proposed the Multiscale Finite Element Method (MsFEM). et al. [3,4], provides an effective mean to reduce the solution scale of FE

analysis. The MsFEM's fundamental concept is to build a link between through the use of numerical base functions, which enable the Small-scale DOFs can be compressed by large-scale ones, so that The scale of the solution can be significantly reduced. Ms. FEM already has been utilized in a variety of heat-related heterogeneous multiscale analyses problems with elastic-plasticity, transfer, and thermal-mechanical coupling problems [7]. Klimczak et al. [8] made the suggestion. a MsFEM-based method for lowering the solution scale in the modeling of heterogeneous viscoelastic materials, which was made even better by adaptive production of both coarse and fine meshes However, this method necessitated iterative computation at each time step. Utilizing the advantages of the polygon element in recent years and Scaled Boundary Finite Element Method, quadtree mesh [9,10] (SBFEM) [11,12] has a tendency to appear in FE form rather than BEM and has been effectively utilized in the multiscale heterogeneous analysis based on FE [13,14]. A multiscale FE2 model was presented by Praser et al. [15] for materials that are hyperelastic and for which quadtree SBFEM is used in the on RVE, numerical homogenization. utilizing SBFEM in conjunction with X. Wang in any way.

MsSBFEM was proposed by Egger [16] to deal with the two-dimensional

linear elastic crack propagation problems, authors developed a polygon and quadtree elements based MsSBFEM in the heterogeneous

analysis for the steady state heat conduction problem [17]. The singular

scaled boundary finite element inherits the advantage of SBFEM to

tackle with the stress/heat flux singularity [18–21], and the polygon

element with arbitrary number of edges and vertices provides convenience to handle irregular geometric shape [9]. By combining with

quadtree gridding techniques, the SBFE mesh generation becomes more

efficient, and can even be directly conducted on the digital images of

computing domains [22,23].

By incorporating the MsSBFEM's advantages and Temporarily

Piecewise Adaptive Algorithm (TPAA) that is able to secure a stable and

high-fidelity temporal solution accuracy when the step size varies

[24–28], a new reduced order approach, i.e., Multiscale Scaled Boundary Finite Element Method (TPA-MsSBFEM), is put forward in the heterogeneous viscoelastic analysis.

To some extent the presented work can be considered as an extension

of our previous work [17], like how linear elastic MsFEM was extended problems with thermal mechanical coupling and elastic-plastic problems [29] problems with viscoelasticity [8] and [30]. One of the most important or significant The novelty of these extensions is the creation of a suitable platform on which MsFEM can be used in the linear elastic case. This means that a The ability to construct a recursive SBFEM on the basis of framework on which MsSBFEM can be used in a linear fashion elastic case, the coarse element stiffness matrices must be evaluated. using numerical base functions to generate only once without iteration is required across the entire time domain. There are numerous advantages in in terms of reducing the solution scale and making dealing with the stress singularity and the production of meshes from complex heterogeneous distribution of the materials and accuracy of the stable temporal solution as well. The following is how the paper is laid out. The recursive is covered in Section 2. viscoelastic problem governing and constitutive equations; Section 3 explains the process of building numerical base functions and two a variety of SBFE gridding methods; Section 4 discusses adaptive recursive multiscale solutions at both small and large scales; and Section 5 demonstrates the efficacy of the proposed method through three numerical examples. examples, and the conclusions and discussions are provided in Section 6.

2. Recursive governing and constitutive equations

2.1. Governing and recursive governing equations

The governing equations of viscoelastic problem can be described by

where {σ}, {ε} denote the vectors of stress and strain, respectively, {b} is

the vector of the body force, {u} is the vector of displacement, and Ω

stands for the domain of interest, and is enclosed by Γ =Γu∪Γσ.

The boundary conditions are specified by

{u} = {u}on Γu (4)

[n]{σ} = {p} on Γσ (5)

where {u} and {p} are the vectors of prescribed displacements and

tractions along the boundaries of Γu and Γσ, respectively, and [n] refers

to the matrix of unit outward normal along Γσ.

We divide time domain into a number of time intervals, the initial

points and sizes of time intervals are defined by t1,t2,..., tk... andT1,T2,...,

Tk..., respectively. At a discretized time interval, in order to describe the

variation of variables more precisely, all variables are expanded in term

of s

where tk-1 and Tk represent the initial point and size of the k-th time

interval, {σm}, {εm}, {bm}, {um}, {pm}, {um}, and {pm} denote the

expanding coefficients of {σ(t)}, {ε(t)}, {b(t)}, {u(t)}, {u(t)} and

{p(t)}, respectively.

With regard to t, the conversion relationship between the differentiations also, s is

Substituting Eq. (6) - Eq. (12) into Eq. (1) - Eq. (5) and equating the

power of two sides of the equation then yields

[H]

σm} + {bm} = 0 in Ω (17)

{εm} = [H]{um} (18)

[n]{σm} = {pm}on Γσ (19)

{um} = {um}on Γu ( 20)

3.2. Polygon and quadtree gridding technique

3.2.1. Polygon mesh

An arbitrary number of polygon elements are utilized in the polygon mesh.

sides and nodes on side [9], which brings convenience for characterizing

complex geometries in the multiscale analysis, as the polygon mesh can

be gridded by only one or very few SBFEs for the region with the same

material property. Fig. 4 shows the polygon gridding for the heterogeneous plate which contains unit cells composed of two kinds materials

represented by different colors, in which a SBFE mesh with only 5

polygon SBFEs can be observed at the small-scale. In addition, it is more

important that this kind of mesh provides a flexible way to increase

nodes of coarse element without adding any fine mesh nodes. As shown

in Fig. 4, on the basis of the commonly used four-node coarse element, a

multi-node coarse element can be constructed by adding any number of

coarse grid nodes (denoted as red color) at any position along the side of

coarse grid.

3.2.2. Quadtree mesh

The quadtree gridding provides an effective tool of the mesh generation for the image-based analysis. In Fig. 5, each pixel in the image is

represented as a square domain which is divided into a number of elements, and all the color densities of the pixels are recorded. If the difference between the maximum and minimum color intensity in an

element is larger than the color threshold, the element is recursively

divided further into four equal-sized elements until all the elements

satisfy the criterion of homogeneity or reaches the minimum edge length

[25]. When a 2:1 rule is used in the process of quadtree gridding in

two-dimensional case, 16 possible element patterns have been provided

for a convenient generation of SBFE stiffness matrix [25]. Besides, the

‘hanging nodes’ produced in the match of adjacent elements in the

quadtree mesh generation can be easily treated as nodes of a new

polygon element, instead of regular ones, as shown in Fig. 5.

Fig. 16. The curves ofstress with distance to crack tip in feature coarse element I

color) will be used to test the influence of the type of coarse element.

Firstly, the computational accuracy is verified by the reference solutions provided by ABAQUS based on FE solution (6669 nodes) for

entire structure shown in Fig. 8. Table 2 shows the variations of

displacement at large-scale feature point 2 with different sizes of time

steps, in which the error tolerance is β= 10-6, the material parameter is

Case 2 and Model C is selected as the coarse element. Also, the

displacement of feature point iii (shown in Fig. 7) in the coarse element

II (shown in Fig. 6(b)) at small-scale is illustrated in Table 3. It is shown

that the maximum eu

i is 0.95% in large-scale and 0.44% in small-scale,

respectively, and the proposed method has a stable accuracy both in

large and small scales when the time step increases from 0.001 s to 0.1 s.

Fig. 9 shows the variation of displacement with time for the feature

point 2 at large-scale and the feature point iii in the coarse element II at

small-scale. It is demonstrated that the proposed algorithm can ensure

the computational accuracy with different time steps both in large and

small scales, while the solutions from ABAQUS change relatively larger

as the time step increase from 0.001 s to 0.1 s.

The variations of the number of recursion steps in time domain is

shown in Fig. 10. As the β changes from 10− 6 to 10− 12 and the size of

time step changes from 0.5 s to 0.1 s, the proposed algorithm can

adaptively adjust the number of recursive steps according to the prescribed error tolerance and the size of time step to maintain computational accuracy.

Secondly, the influence of material heterogeneity for the result is

investigated. Table 4 provides the comparison of proposed algorithm

and reference solution on displacement ux at t= 10s and the corresponding error indicator ψ for all the feature nodes at the large-scale in

Fig. 6(b), respectively. The Model A coarse element is used. The results

for displacements in large-scale with different cases of materials are

shown in Table 1. It is found that as the ρ = EMat.1 2 /EMat.2 2 increases from 2

to 100, the maximum eu

i of a single feature point increase from 2.10% to

4.70%, and ψ increases from 0.0011 to 0.0081. With the increase of the

material heterogeneity, in term of the increase of material ratio, the

solution accuracy of the proposed algorithm decreases gradually.

Thirdly, the influence of the coarse element with different number of