# Numerical upscaling of discrete network models

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## Abstract

In this paper a numerical multiscale method for discrete networks is presented. The method gives an accurate coarse scale representation of the full network by solving sub-network problems. The method is used to solve problems with highly varying connectivity or random network structure, showing optimal order convergence rates with respect to the mesh size of the coarse representation. Moreover, a network model for paper-based materials is presented. The numerical multiscale method is applied to solve problems governed by the presented network model.

## Keywords

Upscaling Multiscale method Discrete network Network model## Mathematics Subject Classification

65N22 34B45## 1 Introduction

Network structures are used to model a wide variety of phenomena, such as flow in porous media, traffic flows, elasticity of materials, body deformation in computer graphics, molecular dynamics, and fiber materials. In these applications, the microscale behaviour determines the macroscale properties of the system. Often a full microscale model is difficult or impossible to work with because of the vast computational complexity. Therefore, there is an interest in constructing coarser, but still accurate, representations of the entire system. Such a procedure is sometimes referred to as upscaling or homogenization. In this work a numerical upscaling method for discrete networks is presented.

There exist several numerical upscaling methods for partial differential equations (PDE) based on the idea of homogenization, such as the heterogeneous multiscale method (HMM) [20], the multiscale finite element method (MsFEM) [8], and the more recent works [3, 17]. The upscaling approach presented in this paper is based on the localized orthogonal decomposition method (LOD) [4, 15], which in turn is inspired by the variational multiscale method (VMM) [9]. Multiscale methods applied to network problems are for instance investigated by Ewing [5] and Ilev et al. [11] who study the heat conductivity of network materials and develop an upscaling method by solving the heat equation locally over small sub-domains. These local solutions are used to compute an effective global thermal conductivity tensor. Della Rossa et al. [2] investigate network models of traffic flows and derive a governing PDE for the macroscale by formulating traffic flow equations for single network nodes and interpreting the relations as finite difference approximations. The macroscale parameters are resolved using a two-scale averaging technique. Chu et al. [1] develop a multiscale method for networks representing flows in a porous medium. The medium is modelled as a network where nodes represent pores and edges represent throats. The conductance of each throat is assumed to be given by Hagen–Poiseuille equation, and using mass conservation equations for the flow through the network, a model for the microscale is attained.

The numerical upscaling method proposed in this work is developed for general unstructured networks. The network is supposed to represent the microscale, and the macroscale is represented by a finite element mesh which is coarse in comparison to the fine scale network. The coarse grid does not have to be related to the network in any way except that both cover the same computational domain, and therefore the method can be applied to arbitrary network geometries. The coarse FEM grid is used to define a macroscale solution space spanned by basis functions defined at each coarse grid node as in standard FEM. The upscaling idea is to modify the coarse basis functions to account for the microscale features of the network. This is accomplished by solving local sub-network problems at each coarse basis function. The modified basis functions are thereafter used to solve a global low-dimensional system resulting in an accurate macroscale solution. The method leads to modified basis functions that decay exponentially, and hence localization of the local sub-network problems can be utilized, reducing the computational cost considerably while preserving optimal convergence rates.

Moreover, this paper includes a two-dimensional network model, which can be used to model paper-based materials in form of fiber networks. The macroscale mechanical properties of paper-based materials are of great interest. Paper is a heterogeneous material built up of fibers bonded together into a network structure. The mechanical properties of paper depend primarily on the properties of the fibers and the bonds between them. In [12, 14, 19], computational fluid dynamics and advanced contact modeling are used to simulate the paper forming process. One future aim is to utilize that framework together with the proposed multiscale method to create virtual fiber networks and investigate the macroscale mechanical properties. A network representation including fibers and bonds is a suitable methodology to study the mechanical properties of paper [10, 13, 18]. Moreover, the varying properties of single fibers and bonds, as well as an interest for fracture propagation simulations, call for an upscaling approach. The presented network model is based on forces arising at the nodes when the network is displaced, acting to restore the initial configuration. The network model is similar to lattices models like [16, 21] where edges are represented by springs. Moreover, angle springs between pair of edges are included. A novelty of the network model in this work is a third type of force phenomenon resulting in an effect similar to the Poisson effect. Force equilibrium equations at each node result in a matrix equation which can be very large. For a regular network, the model converges to the linear elasticity equation when the length of the network edges tends to zero. The numerical upscaling method is applied to the network model and numerical examples are solved to demonstrate the convergence rates of the method. The examples show how the proposed numerical upscaling method resolves fine scale features which the standard FEM cannot.

The outline of this text is as follows. In Sect. 2, the general problem formulation is stated. Thereafter, in Sect. 3, the theory of the numerical upscaling method is presented. Sect. 4 contains error analysis, and in Sect. 5, the two-dimensional network model is described. In Sect. 6, numerical examples are presented, showing the convergence rates of the proposed method. Lastly, in Sect. 7, conclusions and future work are discussed.

## 2 Problem formulation

*K*can for instance be the discrete Poisson operator describing heat conduction, the finite difference discretization of the linear elasticity operator, or represent a more complex model, such as of the mechanics of a fiber network. Let \(F\in \mathbb {R}^{n}\) denote the load vector and let the solution vector be denoted

*u*, belonging to a vector space \(V\subset \mathbb {R}^n\). The network problem can be stated in two equivalent ways, either:

*K*and

*F*by explicitly including the restriction of

*u*to the space

*V*, for instance by holding some nodes fixed. To ensure existence and uniqueness of the second formulation, (2.2), it is assumed that

*K*is symmetric and positive definite on

*V*. A matrix \(K\in \mathbb {R}^{n\times n}\) is positive definite on a subset \(V\subset \mathbb {R}^n\) if \(v^TKv>0\) for all nonzero \(v\in V\). Moreover, a symmetric positive definite matrix

*K*constitutes a scalar product \(\langle u,v\rangle =u^TKv\) on

*V*, a property that will be used later.

*K*as the resulting discretization of the linear elasticity operator. This problem setup can be used to find the node displacements

*u*under applied node forces

*F*. To attain a solvable system \(Ku=F\), some degrees of freedom have to be prescribed, resulting in the restricted solution space

*V*. The network in Fig. 1b can represent a conductive medium, governed by the discrete Poisson equation. The temperature at each node is contained in

*u*. The network in Fig. 1c is a fiber network building up a paper sheet. The fibers are modelled as chains of edges connected at nodes with bonds between fibers at common network nodes.

## 3 Numerical homogenization of networks

Consider a network with *N* nodes and properties governed by a symmetric and positive semi-definite matrix \(K\in \mathbb {R}^{n\times n}\), where \(n=d\cdot N\) is the number of degrees of freedom of the network, and *d* denotes the number of degrees of freedom at each node. For instance, for an elastic network, where node displacements are to be solved, the number of degrees of freedom at each node will be two or three, depending on if the network is two- or three-dimensional. In the following presentation the space is assumed to be two-dimensional, but the method works analogously for three dimensions. Denote by \(p_i\in \mathbb {R}^2\) the position of the node corresponding to degree of freedom \(i = 1, \dots , n\). Note that groups of *d* degrees of freedom correspond to the same position.

*u*(1) and

*u*(2) correspond to the first and second degree of freedom of node 1,

*u*(3) and

*u*(4) correspond to the first and second degree of freedom of node 2, and so on, with analogous ordering if

*d*is larger. Here

*u*(

*i*) denotes the

*i*:th component of vector

*u*. Let \(F\in \mathbb {R}^{n}\) denote the load vector. The system \(Ku=F\) is not necessarily solvable without prescribing some degrees of freedom. Consider fixed constraints with zero displacement (non-zero displacement is treated in Sect. 3.4) and let \(\mathscr {N}_D\subset \{1,\dots , n\}\) be the set of indices corresponding to the fixed degrees of freedom. Let \(\mathscr {N}=\{1,\dots , n\}{\setminus }\mathscr {N}_D\). Denote by \(V \subset \mathbb {R}^{n}\) the restricted solution space defined by

*K*in addition to being symmetric, also is positive definite on the restricted solution space

*V*.

### 3.1 Coarse grid representation

*d*number of basis functions are defined similarly as in the finite element method. These basis functions span a low dimensional solution space which gives an insufficient description of the fine scale features. To include the fine scale information, the basis functions are modified by solving local sub-network systems. Thereafter the modified basis functions are used to solve a global system, smaller than the full system including all nodes, resulting in an upscaled approximation of the original problem. In what follows, the details of this procedure are described.

Let the coarse grid be denoted \(\mathscr {T}\), containing *M* coarse nodes and let \(m=d\cdot M\) be the degrees of freedom of the coarse grid. One choice of coarse grid is a quadrilateration as in Fig. 2a. It is assumed that the coarse grid constitutes a good approximation of the computational domain of the network, and that each coarse element contains at least one network node and that \(N> M\). Let \(\varLambda _i:\mathbb {R}^2\rightarrow \mathbb {R}, \,i=1,\dots , m\), denote the coarse nodal basis functions of the grid \(\mathscr {T}\). For a quadrilateration, bilinear basis functions are suitable, illustrated in Fig. 2b.

*x*-displacement of that node. If there exists a network node with fixed degree of freedom

*j*such that \(p_j\) lies in the support of \(\varLambda _i\), and

*j*also describes

*x*-displacement, then the coarse degree of freedom

*i*should be fixed. This is illustrated in Fig. 3. The fixation condition is equivalently stated as:

*i*and

*j*, correspond to the same displacement direction (e.g.

*x*or

*y*for \(d=2\)). For networks with more complex boundary geometry, the coarse grid has to be refined at the boundary to attain a proper representation of the fixed boundary conditions.

Let \(\mathscr {M}=\{1, \dots , m\}{\setminus } \mathscr {M}_D\) denote the set of nonprescribed coarse degrees of freedom. The positions of the coarse nodes, \(\{P_i\}_{i=1}^m\), defined similarly as the positions of the network nodes, are a subset of \(\mathbb {R}^2\), likewise as the nodes of the network. However, these two subsets do not have to be related, but as already noted, it is assumed that each coarse element contains at least one network node.

*W*. The coarse space is defined from the coarse basis functions in the following way. Let \(\lambda _i\in \mathbb {R}^n, i=1,\dots , m\), be the interpolation of the coarse nodal basis functions to the network nodes given by

*v*is zero for each interpolated bilinear basis function \(\lambda _i\), i.e. \(\lambda _i^T v= 0,\,\forall i\in \mathscr {M}\).

*V*such that each \(v\in V\) can be uniquely decomposed as \(v= v_H+w\) where \(v_H\in V_H\) and \(w\in W\). Before proving this fact, a lemma is stated showing the relation between the spaces \(\mathbb {R}^{m_H}\), \(\mathbb {R}^{n}\) and \(V_H\), and the maps in-between, illustrated in Fig. 4.

### Lemma 3.1

If \(B_H\) has linearly independent columns and \(C_H=B_H^T\), then for each \(v_H\in V_H\) there exists \({\bar{v}}_H\in V_H\) such that \(B_HC_H{\bar{v}}_H = v_H\). Moreover, if \(v_1, v_2\in V_H\) such that \(B_HC_Hv_1 = B_HC_H v_2\), then \(v_1=v_2\).

### Proof

The map \(B_H:\mathbb {R}^{m_H}\rightarrow V_H\) is one-to-one since \(B_H\) has linearly independent columns. Moreover, the map \(C_HB_H:\mathbb {R}^{m_H}\rightarrow \mathbb {R}^{m_H}\) is invertible since \(C_HB_H=B_H^TB_H\) is symmetric and positive definite due to the fact that \(x^TC_HB_Hx=|B_Hx|^2 \ge 0\) and \(|B_Hx|=0\) implies \(x=0\). Given \(v_H\in V_H\), it exists \(a\in \mathbb {R}^{m_H}\) such that \(B_Ha=v_H\), and since \(C_HB_H\) is invertible it exists \(b\in \mathbb {R}^{m_H}\) such that \(C_HB_Hb=a\). Therefore \(v_H=B_HC_HB_Hb\) leading to \({\bar{v}}_H = B_Hb\in V_H\).

To prove the second part, assume \(v_1\ne v_2\). Then there exists \(a_1,a_2\in \mathbb {R}^{m_H}\) with \(a_1\ne a_2\) such that \(v_1=B_Ha_1\) and \(v_2=B_Ha_2\). Since \(C_HB_H\) is invertible, \(C_HB_Ha_1\ne C_HB_Ha_2\), contradicting the fact that \(B_HC_HB_Ha_1= B_HC_HB_Ha_2\), hence \(v_1=v_2\). \(\square \)

### Proposition 3.1

If \(B_H\) has linearly independent columns and \(C_H=B_H^T\), then \(V = V_H\oplus W\) uniquely.

### Proof

For \(v\in V\), let \(v_H=B_HC_Hv\). Lemma 3.1 states the existence of \({\tilde{v}}_H\in V_H\) such that \(v_H=B_HC_H\tilde{v}_H\). Since \(B_H\) has linearly independent columns the relation \(0=v_H-v_H=B_HC_Hv-B_HC_H{\tilde{v}}_H = B_H(C_Hv-C_H{\tilde{v}}_H)\) implies that \(C_Hv-C_H{\tilde{v}}_H = 0\) with conclusion that \(v-\tilde{v}_H\in W\). Therefore \(v={\tilde{v}}_H+(v-{\tilde{v}}_H)\) is a desired decomposition. To show uniqueness, consider two decompositions \(v=v_{H,1} + w_1\) and \(v= v_{H,2}+w_2\). Then \(v_{H,1} + w_1= v_{H,2}+w_2\), and applying \(B_HC_H\) on both sides gives \(B_HC_Hv_{H,1}=B_HC_Hv_{H,2}\). From the last part of Lemma 3.1 it follows that \(v_{H,1}=v_{H,2}\). \(\square \)

*W*, together with the connectivity matrix

*K*, the multiscale space \(V_\mathrm{ms}\) is defined as the

*K*-orthogonal complement of

*W*:

*W*and \(V_\mathrm{ms}\) constitute another splitting of

*V*implying that every \(v\in V\) can be decomposed uniquely as \(v = v_\mathrm{ms}+w\) where \(v_\mathrm{ms}\in V_\mathrm{ms}\) and \(w\in W\).

### Proposition 3.2

Assume \(B_H\) has linearly independent columns and \(C_H=B_H^T\). If *K* is symmetric and positive definite on *V*, then \(V = V_\mathrm{ms}\oplus W\) uniquely.

### Proof

Consider \(v\in V\). From Proposition 3.1 it is known that \(v=v_H+ {\tilde{w}}\) with \(v_H\in V_H\) and \({\tilde{w}}\in W\). Let \(z\in W: \, w^TKz=w^TKv_H,\, \forall w\in W\), which has a unique solution since *K* is symmetric and positive definite on *V*. Define \(v_\mathrm{ms}=v_H-z\) and \(w={\tilde{w}} + z\), where the second sum is in *W*. Since \(x^TKv_\mathrm{ms}=x^TKv_H - x^TKz=0, \,\forall x\in W\), it is true that \(v_\mathrm{ms}\in V_\mathrm{ms}\), giving the desired decomposition as \(v = v_\mathrm{ms} + w\). To prove uniqueness, consider \(v=v_{\text {ms}, 1} + w_ 1\) and \(v=v_{\text {ms}, 2} + w_ 2\). Then \(0=x^TK(v-v) = x^TK(w_1 + v_{\text {ms}, 1} - w_2 - v_{\text {ms}, 2})= x^TK(w_1-w_2), \, \forall x\in W\), implying that \(w_1=w_2\). \(\square \)

### Proposition 3.3

Let *K* be symmetric and positive definite on *V*, and \(F\in V\). Then there exists a unique solution to problem (3.3).

### Proposition 3.4

Let \(u_f\in W\) be such that \(w^TKu_f=w^TF, \,\, \forall w \in W\). Then the sum \( u = u_\mathrm{ms} + u_f\), where \(u_\mathrm{ms}\) is the solution to the multiscale problem (3.3), solves the original problem (3.1).

### Proof

### Proposition 3.5

If *K* is symmetric and positive definite on *V*, \(B_H\) has linearly independent columns and \(C_H=B_H^T\), then the vectors \(\{\lambda _i-\phi _i\}_{i\in \mathscr {M}}\) constitute a basis for \(V_\mathrm{ms}\).

### Proof

The problem to find \(\phi _i\) such that \(w^TK\phi _i=w^TK\lambda _i\), \(\forall w\in W\), has a unique solution \(\phi _i\in W\) since *K* is symmetric and positive definite on *V*. By construction, it is also true that \(\lambda _i-\phi _i\in V_\mathrm{ms}\). To prove linear independence, consider \(\sum a_i (\lambda _i-\phi _i)=0\) and apply \(B_HC_H\) to both sides. Using that \(C_H\phi _i=0\) gives \(B_HC_H\sum a_i\lambda _i = 0 = B_HC_H0\), and by the second part of Lemma 3.1 it follows that \(\sum a_i\lambda _i = 0\) implying \(a_i=0\). \(\square \)

At this point, a multiscale space with low dimension compared to the solution space *V* has been constructed as was desired in the problem formulation. Moreover, it has been shown that the problem can be solved through a matrix equation after constructing a basis for the multiscale space. However, the problems of solving the modified basis functions have the same size as the original problem. It turns out that this can be circumvented, utilizing that the modifications \(\phi _i\) decay exponentially, implying that the problems can be localized. This is presented in the following section.

### 3.2 Localization

The described method requires systems to be solved which are as large as *K*. However, as will be demonstrated by numerical examples in Sect. 6, the modifications \(\phi _i\) decay fast away from its coarse node. Therefore the problems (3.4), of calculating \(\phi _i\), can be localized with preserved convergence rates. The localization is accomplished by solving each problem (3.4) on a restricted domain, called patch.

*K*over each coarse element

*E*, such that \(K=\sum _{E\in \mathscr {T}} K_E\). The local element connectivity matrices \(K_E:V\rightarrow V\) are assembled for each element \(E\in \mathscr {T}\) by only considering edges in each element. See Fig. 5 for an illustration. For unstructured networks, edges may intersect the elements. Such a situation is resolved by temporarily dividing the edges at intersection points. Using the decomposition of the connectivity matrix into local element matrices, the modifications \(\phi _i\) can analogously be assembled as \( \sum _{E\in \mathscr {T}} \phi _i^E\), where \(\phi _i^E\) is the solution to the problem

### Proposition 3.6

The sum of the elementwise modifications, \( \sum _{E\in \mathscr {T}} \phi _i^E\), solves the original problem (3.4).

### Proof

*E*, as depicted in Fig. 6b. Let the patch size be described by the parameter \(\rho \) such that \(\rho H\) is the radius of the patch, where

*H*denotes the coarse element size. In the two examples shown in Fig. 6, the value of \(\rho \) corresponds to 1.5. Let \(\mathscr {N}_E \subset \mathscr {N}\), denote the degrees of freedom of network nodes that are in the patch \(\omega _E\). Similarly, let \(\mathscr {M}_E \subset \mathscr {M}\), denote the degrees of freedom of coarse nodes that are in the patch of element

*E*.

*E*, its localized subspace of the detail space is defined according to

*E*, and taking the sum \( {\tilde{\phi }}_i = \sum _{E\in \mathscr {T}} {\tilde{\phi }}_i^E\). The resulting localized multiscale space is denoted \({\tilde{V}}_\mathrm{ms}\), and is defined as

### 3.3 Algebraic formulation

In this section the algebraic formulation of the numerical multiscale method is presented. The multiscale method consists of two main steps. First, the basis for the multiscale space \(V^\text {ms}\) is constructed by calculating each \(\phi _i\) from (3.4). Secondly, the attained modified basis is used to solve the global multiscale problem (3.3). In this section, elementwise assembling and localization, described in Sect. 3.2, is employed.

First, a useful matrix notation is introduced. Consider a matrix \(A\in \mathbb {R}^{a\times b}\). Let \(\mathscr {A}\subset \{1,2,\dots ,a\}\) and \(\mathscr {B}\subset \{1,2,\dots ,b\}\) be subsets of the matrix row and column indices respectively. A new matrix \(A(\mathscr {A}, \mathscr {B})\in \mathbb {R}^{|\mathscr {A}|\times |\mathscr {B}|}\) is extracted from *A* by only considering rows corresponding to indices in \(\mathscr {A}\) and columns corresponding to indices in \(\mathscr {B}\), that is \(A(\mathscr {A}, \mathscr {B})= (a_{ij})_{(i,j)\in \mathscr {A}\times \mathscr {B}}\) .

### 3.4 Non-zero fixed boundary conditions

## 4 Error analysis

*K*and a right hand side load

*F*. In this section, some error bounds are shown. Because of the generality of

*K*, assumptions are needed. It is assumed that the coarse grid is a quasi-uniform finite element mesh with mesh parameter \(H\approx m^{-1/2}\). The following two norms on the space

*V*are introduced:

*i*. It is assumed that

*K*is symmetric and positive definite on

*V*, and that the smallest eigenvalue of

*K*, denoted \(\nu \), fulfils

*n*. For the finite element method posed on a quasi-uniform mesh these definitions and assumptions correspond to the energy norm, a weighted \(l^2\)-norm and a Poincaré inequality. A bilinear weighted interpolant \(\pi _H :V\rightarrow V_H\) is defined as \(\pi _H v=\sum _{i\in \mathscr {M}}(\lambda _i^T v)\lambda _i\). i.e. \(\pi _H v = B_HC_Hv\). Assume the following error bound:

*O*(1) in regard to

*n*for distributed surface loads.

*C*is independent of data variation in the connectivity matrix, and without loss of generality it is assumed that \(F\in V_H\).

First it is shown that the multiscale solution \(u_\mathrm{ms}\) is equal to *u* if \(F\in V_H\).

### Proposition 4.1

*u*and \(u_\mathrm{ms}\) be defined according to

### Proof

*W*. Therefore \(u_\mathrm{ms}\) solves the same equation as

*u*, and due to uniqueness \(u_\mathrm{ms}=u\). \(\square \)

*K*. It has however been analysed for several concrete cases, for instance when

*K*is the stiffness matrix arising from discretizing the Poisson equation [15] and the elasticity equations [7] with the finite element method. For these cases it is true that

*C*and

*c*are constants independent of

*H*. In Sect. 6 it is shown, through numerical validation, that this relation is true for the more complicated network model considered there. The theoretical analysis of this result for the general network model proposed in Sect. 5 is complicated and postponed to future work. Assuming this result gives the following error bound.

### Theorem 4.1

### Proof

## 5 Network model for paper-based materials

In this section, a two-dimensional elasticity network model is presented, which can be used to model fiber networks in paper-based materials. An elasticity network consists of nodes and edges. When the nodes are displaced, internal forces act to restore the displacements. These forces act at two types of elements, either on edges or on edge pairs (two edges connected at a joint node). Three types of internal forces are included in this model, one type is related to edges, and two types are related to edge pairs. The model is two-dimensional, static and assumes small deformations.

*i*reads

Let the network consist of *N* nodes. Let (*i*, *j*) denote the edge connecting node *i* to *j* and let \(\mathscr {E}\) denote the set of all edges. Note that \((i, j) = (j, i)\). Edge pairs are denoted by (*i*, *j*, *l*) where *j* is the central node. Denote by \(\mathscr {P}\) the set of all edge pairs. Note that \((i,j,l)=(l,j,i)\). Each node *i* has two degrees of freedom, the *x*-directed displacement and the *y*-directed displacement, contained in the vector \(\delta _i\in \mathbb {R}^2\).

*K*is attained by summation of matrices assembled at edges and edge pairs. The node displacement vector

*u*is arranged according to

*i*is at row \(2i-1\) for the

*x*-component, and at row 2

*i*for the

*y*-component.

*i*,

*j*) be denoted \(L_{ij}\) and assume that the edge has a width \(w_{ij}\). All edges is assumed to have a uniform thickness

*z*in the direction into the plane. The direction vector, \(d_{ij}^a\), of an edge (

*i*,

*j*), with respect to node \(a\in \{i,j\}\), is defined as

*i*,

*j*) is denoted \(\varDelta L_{ij}\) and given by

### 5.1 Extension of edges

The first force contribution acts at edges due to their internal resistance to length change. When the nodes of an edge are displaced so that the projection of the difference of the node displacements onto the initial edge direction is nonzero, anti-parallel forces arise at the nodes of the edge to restore the length. The tendency of an edge to restore its length is described by the elastic modulus \(k_{ij}\).

*i*,

*j*) as shown in Fig. 7b. When the nodes are displaced, the edge (

*i*,

*j*) will give rise to two forces \(F_a^\text {I}(i,j)\), \(a\in \{i,j\}\), acting on node

*a*according to

### 5.2 Angular deviations of edge pairs

The second force contribution acts at edge pairs from their internal tendency to resist change of the angle between their two edges. When a change in angle occurs, two torques arise at the connecting node acting on one edge each to restore the change. By transforming these torques to force couples the effect can be converted into the force equilibrium equations.

*i*,

*j*,

*l*) as depicted in Fig. 7a. When the nodes are displaced, an angular change \({\varDelta \theta }_{ijl}\) occurs, giving rise to two torques

*i*,

*j*) and (

*j*,

*l*) respectively, at the position of node

*j*. The angular change is a sum of two contributions according to

*i*,

*j*,

*l*), acting on node

*a*, according to

### 5.3 Poisson effect of edge pairs

The third force contribution results in an effect similar to the Poisson effect and acts at edge pairs. The idea is to add forces that work to keep the total length of the two edges of the pair constant. Hence, when one edge changes length, two kind of forces occur, on one hand forces acting to restore the length of the specific edge, on the other hand forces acting to change the length of the other edge in the pair.

*i*,

*j*,

*l*), as shown in Fig. 7a. The forces acting at the outer nodes \(a\in \{i,l\}\) will be

### 5.4 Assembly of elasticity matrix

*K*by explicitly including boundary conditions. Since each edge and each edge pair leads to separate force contributions, the total elasticity matrix

*K*can be assembled from separate element matrices for each of the three force contributions. Let \(K^\text {I}_{ij}, K^\text {II}_{ijl}, K^\text {III}_{ijl}\in \mathbb {R}^{n\times n}\) denote the matrices assembled from the first, second and third force contribution respectively, at different elements [edges (

*i*,

*j*) or edge pairs (

*i*,

*j*,

*l*)]. These matrices are sparse and the only nonzero elements are defined by the relations

*K*is assembled according to

*K*is symmetric and semi-positive definite. With proper fixation of nodes, the matrix will be positive definite on the restricted solution space. Moreover, for a regular network with uniform coefficients

*k*, \(\kappa \), \(\eta \) and \(\gamma \), the presented model is equivalent to the finite difference discretization of the two-dimensional linear elasticity equations.

## 6 Numerical results

*i*. Set the parameters of the network model to

*i*,

*j*). Using the above parameters, the elasticity matrix

*K*is assembled. Let \(h=1/r\) denote the length of each network edge.

*F*by explicitly including the boundary conditions.

*x*-directed degrees of freedom with \(x=1\).

To solve the two problems using the proposed numerical multiscale method, a coarse FEM grid is introduced. The grid is similar to the network but with \((R+1)^2\) nodes. The basis functions \(\varLambda _i\) are chosen as classic bilinear. See Fig. 9b for an illustration of the network and coarse FEM grid. Let \(H= 1/R\) be the width of the coarse elements. With the described network geometry, the fixed boundary conditions correspond to fixation of coarse nodes at the boundary.

*i*is displaced \([\delta x_i,\, \delta y_i]\) where \(\delta x_i\) and \(\delta y_i\) is randomly sampled in \([-0.4h, 0.4h]\) with uniform distribution. For nodes with initial coordinate \(x=0\) or \(x=1\), it is enforced that \(\delta x_i = 0\), and similarly for nodes with \(y=0\) or \(y=1\), \(\delta y_i=0\). In Fig. 10, a network with such random structure is shown for \(r=32\).

*E*. Hence patches will have the circular form as was illustrated in Fig. 6a. The rapid decay of the modified basis \(\{\lambda _i-\phi _i\}_{i\in \mathscr {M}}\) is demonstrated by solving the modification \({\tilde{\phi }}_i\) with different \(\rho \) and computing the relative error \(\vert \vert \vert \phi _i-{\tilde{\phi }}_i\vert \vert \vert /\vert \vert \vert \phi _i\vert \vert \vert \). The degree of freedom

*i*is chosen as one of the central nodes but the trend is similar for all nodes. The resulting errors for the first problem (6.1) are seen in Fig. 11.

*D*denotes the diagonal matrix with values \({\bar{h}}_i^2, \,\, i=1, \dots , n\) on the diagonal. This is done for the three different setups and \( r \in \{2^2, 2^3, 2^4, 2^5, 2^6\}\). In Fig. 12, it is seen that the resulting eigenvalues are independent of

*n*. The values are normalized with the value for \(r=2^6\).

## 7 Conclusion and future work

In this paper a numerical multiscale method for discrete networks is proposed. For a set of different numerical examples, the convergence rates of the proposed method are examined. For regular networks with low connectivity variation the method resembles the convergence rates of the ordinary FEM. For networks with randomly varying connectivity it is shown that the multiscale method performs better than ordinary FEM. The method is moreover used to solve network problems with random structure, indicating error convergence rates at least linear in energy norm and quadratic in \(l^2\)-norm.

A challenging theoretical problem for the future is to extend the result in (4.5) beyond finite element based discretization to more general networks, including the presented network model. Further on, other fiber network models, for example beam based, as well as three-dimensional, should be considered. Thereby the presented numerical upscaling method will be applied to realistic macroscale paper networks, investigating the problem size which can be studied and the computational efficiency of the proposed method. Moreover, the proposed method will be used together with the paper forming simulation framework presented in [12, 14, 19] to study virtual paper sheets and macroscale mechanical properties such as tensile strength, tensile stiffness, bending resistance, *z*-strength and fracture propagation.

## Notes

### Acknowledgements

This work is a part of the ISOP (Innovative Simulation of Paper) project which is performed by a consortium consisting of Albany International, Stora Enso and Fraunhofer-Chalmers Centre. The second author was partially supported by the Göran Gustafsson Foundation.

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