Enhanced Upward Translations for Systems with Clusters

Abstract

The paper is concerned with using boundary element methods (BEM) for the accurate evaluation of fields in structures with clusters. For large-scale problems, the BEM system is solved iteratively by speeding up matrix-to-vector multiplications by applying a kernel-independent fast multipole method. Multiplication starts with source-to-multipole (S2M) translations, whose accuracy predefines the overall accuracy. We aim to increase the accuracy of these translations. The intensities of sources are assembled into clusters by an algorithm suggested. Each of them is characterized by its representative source, whose intensity equals the sum of the intensities of cluster sources. Thus, with growing distance, its field tends toward the field of the cluster. The accuracy of S2M translations is increased by subtracting from and adding to the far field of the cluster the far field of its representative source, and by using the proposed modified kernel to evaluate the difference of the fields, which decreases faster than the field of the cluster itself. Numerical results for typical kernels show a notable increase in the accuracy provided by the modified S2M translations. Keeping in them merely the added field is acceptable for many practical applications. This simplifies the modified S2M translations by avoiding calculation and storing matrices specific to each of the clusters. The improved translations may be also used for multipole-to-multipole translations, performed on next, after leaves, levels in upward running a hierarchical tree.

1. Introduction

A wide range of practical problems, concerning strength and safety, require accurate evaluation of local fields. Such are problems of rock and fracture mechanics, for which extreme, rather than average, values of physical fields are of prime significance (e.g., [1,2,3,4,5,6]). These problems are characterized by a strongly inhomogeneous distribution of flaws, in particular, faults, cracks, inclusions of various sizes and mutual location. Individual flaws of large sizes, and collections of closely located flaws, are normally the sources of the most dangerous events. Such flaws may be related to clusters. They strongly influence not only the safety of engineering constructions but also the accuracy of their numerical modeling. The present paper tends to suggest a means to maintain accuracy. In view of the mentioned applications, we shall have in mind, first of all, static 2D and 3D problems involving Laplace and Navier equations.

When accounting for local fields in inhomogeneous structures with elements interacting on multiple contacts, the boundary integral equations (BIE) appear to be an appropriate means (e.g., [7,8,9,10,11,12]). Their special forms reduce a mathematical problem to boundaries of structural elements and contain as densities the very quantities which describe contact conditions. The BIEs are solved by the boundary element methods (BEM), which employ discretization of boundaries onto a set of boundary elements with approximation densities at each of them (e.g., [8,9,10,11,12]). The main drawback of a BEM is that the matrix of its interaction coefficients is fully populated. Consequently, each of the repeatedly used operations of matrix-to-vector multiplications has the complexity $N^2$. With growing $N$, the time expense becomes enormous. Nowadays, this drawback is smoothed by employing specially designed methods. They decrease the complexity of matrix-to-vector multiplication to the level usual for finite element methods (e.g., [13,14,15,16,17,18,19,20,21,22,23,24,25]).

The first such method was suggested by Rokhlin [13]. It employs a priori geometrical information on element locations, and the analytical expression for an integral kernel. Details of the method called the analytical fast multipole method (A-FMM), may be found in the textbook [19]. Of essence, it employs the geometrical data to build the hierarchical tree. The analytical expression for a kernel is used for its multipole expansion and for organizing translations of fields in up-downward runs over the tree. Thus, the method is kernel dependent. It performs the matrix-to-vector multiplication with the computing complexity to at most $O(N logN)$.

The other breakthrough method is the adaptive cross approximation (ACA) algorithm by Bebendorf [16]. In contrast with the A-FMM, the ACA algorithm is purely algebraic. It is independent of the underlying formulation, in particular, on the knowledge of integral kernel and geometrical features of a physical system. The latter are accounted for implicitly by appropriate algebraic manipulations. In applications to static and low-frequency problems involving asymptotically smooth (non-oscillatory) kernels, the ACA algorithm has shown the complexity $O(N logN)$. Later on, it was successfully applied to problems of electromagnetic scatter and radiation (e.g., [26,27]) for which the kernels are less smooth.

Clearly, however, the geometrical information is of great value for speeding calculations in static and low-frequency problems of fluid and solid mechanics. The mentioned problems of rock and fracture mechanics are among them. It is reasonable to make use of a priori geometrical information in a way that avoids involved analytical work for each particular kernel. The method suggested in the paper [18] was performed in a kernel-independent form. The method follows the same line as the A-FMM in building the tree and up-downward runs over it. The difference is in the design of translations. Instead of using multipole expansions of kernels, the translations are performed by means of integration over specially chosen equivalent surfaces. The integrand is the product of the kernel by the equivalent density. The latter is found by equating the field, corresponding to the integral to the field known on a specially chosen check surface. Details of the method are provided in the textbook [20]. Suggested initially for static and asymptotically smooth kernels, later on, the method has been extended to highly oscillating kernels [28]. The extension employs scaling the problem by wavelength. Since the method does not depend on a particular kernel, the authors called it the kernel-independent fast multipole method (KI-FMM). This terminology was accepted in later papers using and developing the method (e.g., [21,23,24,25,28,29,30]). Henceforth we shall also use the author’s interpretation of the kernel independence although it is notably less general than that provided by the ACA algorithm.

For our theme, two features of the KI-FMM are significant. Firstly, it explicitly employs geometrical data on the mutual location of boundary elements. This makes it possible to evaluate the matrix of distances, which is basic in problems concerning clusters [31]. The problems of rock and fracture mechanics are of the same kind. Secondly, in our paper [24] it is established that there is no need to integrate the product of equivalent density by kernel over the equivalent surface. Instead, we may use equivalent intensities of pointed sources located on the equivalent surfaces. The intensities are promptly found by using checkpoints on the check surface. The interpretation of KI-FMM in terms of equivalent intensities, having clear physical meaning, further extends the options to account for clusters. We shall use both features of the KI-FMM in the next section to give a strict definition of the cluster and to suggest a simple algorithm which rapidly generates the clusters.

The need to use the concept of clusters [31] becomes evident from the numerical results reported in our paper [24]. They show the drastic (in some orders) loss of the accuracy of S2M translations for sources near the equivalent surfaces of leaves. However, there have been no suggestions, or even discussion, of how to improve the accuracy.

Our objective is to numerically distinguish clusters within a leaf and to accurately translate their fields to the check surface of the parent cell. The purpose is reached by using two innovations. The first of them consists of appropriately ordering the leaf sources and grouping them in clusters by using data on their location and intensities. The second consists of introducing and employing modified translations, which translate far fields of each of the clusters more accurately than conventional S2M translations.

Throughout the following discussion, we shall use the concepts and terminology of FMM, explained in detail in the textbooks [19,20]. In particular, the far field is assumed as the combined field, generated by the sources in a tree cell outside the closest neighbors of the cell (e.g., [19,21,28,29,30]). For reasons explained, the exposition is provided for the FMM in its kernel-independent form. The comment on the extension to the analytical FMM is provided in the last paragraph of the paper.

The paper is organized as follows. Section 2 contains guiding considerations for grouping the leaf sources into clusters in accordance with the objective of increasing the accuracy of S2M translations. They yield an algorithm for appropriately distinguishing the clusters. Each of them is characterized by its representative point source. In Section 3, the representative source of a cluster is used to modify S2M translation by means of the “subtract and add” approach. This improves the accuracy of approximate far fields, generated by equivalent sources located at an equivalent surface of the leaf. Numerical examples for typical kernels employed in applications, presented in Section 4, demonstrate the increase in accuracy. In Section 5, it is shown that for many practical applications, the improved modified translations may be simplified by neglecting the input of a residual far field as compared with the input of the representative source. This excludes the need to calculation and store the modified matrices specific to each of the clusters. The results obtained are summarized in Section 6.

2. General Considerations on Clustering as Applied to Upward Translations

Our objective is to increase the accuracy of the starting S2M translations when solving a problem for systems with multiple concentrators of fields, such as crack tips, common vertices of grains, edges of thin inclusions, corner points at boundaries, etc. The areas of high concentration are associated with clusters, and the accuracy of their translated fields influences the accuracy of final results. As known (e.g., [31], https://en.wikipedia.org/wiki/Cluster_analysis, accessed on 15 July 2024), in general, the notion of a “cluster”, cannot be precisely defined. Its exact definition is problem-oriented. Hence, we need to delineate the specific features of our problem, which direct to an exact definition of a cluster and to a corresponding algorithm. These features are as follows.

A typical BEM employs discretization of boundaries and approximation of the densities by form-functions with nodes at assigned points. The nodal points serve to build a hierarchical tree of a FMM. The total far field, generated by elements of a leaf and corresponding to a particular kernel of the starting BIE, may be evaluated as the sum of products of the kernel at a nodal point by an intensity at this point. The intensity itself is defined as the product of the nodal value of the density (scalar or vector) by integral of appropriate form-function over the element containing this node. Thus, the far field of a leaf is presented by the sum of far fields of point sources located in the leaf. The intensities of the sources for quantities, defined by boundary conditions, are known. They are accounted for in the right-hand side of an algebraic system to be solved. The intensities of the remaining sources are unknown. For large systems, they are found iteratively using matrix-to-vector multiplication on each iteration. The matrix in the product consists of influence coefficients, calculated for unit intensities.

The vector of intensities, used in the matrix-to-vector product, is presumed known at the beginning of a current iteration (for the first iteration, rough approximate initial values are found by a simple algorithm, defined, for instance, by a preconditioner). Thus to the beginning of a current iteration performed by an FMM, the intensity and location of each source in a leaf are known. We see that distinguishing a cluster in a leaf may involve merely (i) the intensity of each source in a leaf, and (ii) its location. Knowing locations makes also known the distances between the sources.

Only this information on sources is actually needed and used in S2M translations, which predefine the accuracy of a whole chain of up- and downward translations. As has been established in [24], the accuracy of conventional S2M translations strongly depends on the locations and intensities of sources. We aim to increase the accuracy of translations by constructing and appropriately employing clusters of closely located sources.

A detailed analysis of the accuracy of S2M translations is provided in [24]. It shows that for a leaf with the equivalent surface (ES) of radius $R_e$ (Figure 1a), the smallest error of S2M translations is observed for sources located in the middle of a leaf. The error increases as the sources approach the ES. The greatest error arises for sources located near the ES of a leaf. These are sources in the ring (spherical layer) of radii $0.75R_e$ and $R_e$ (shaded area in Figure 1a). Specifically, as established in the paper [24], the error grows from two orders for weakly singular operators to five orders for singular and hypersingular operators. The greatest error arises for sources located near the ES of a leaf.

Actually, only this group of sources is essential for assembling them into clusters with the aim of increasing the accuracy of translations. Since they are within the layer of the thickness $h=0.25R_e$, the distance $r$ of each of them to the leaf center exceeds $0.75R_e$. This implies that there is no sense in taking the cluster radius $r_{cl}$ exceeding the thickness $h$ of the spherical layer; thus $r_{cl}<0.25R_e$. For the average location of the source centers on the middle surface of the spherical layer, the reasonable value of the cluster radius is $r_{cl}<0.5h=0.125R_e$. The inequality $r>0.75R_e$ may serve to distinguish the sources, for which it is reasonable to improve S2M translation. Thus merely these sources may be assembled into clusters. The remaining leaf sources may be clustered or may stay ungrouped. In the latter case, their far fields are translated conventionally by using S2M translations, described, for instance, in [18,20]. For them, the errors of conventional translations are much less than those for sources in the spherical layer.

In general, a BEM may employ a number of kernels. We consider any of them, and further discussion refers merely to those leaf sources, whose far fields are generated by a particular kernel $\mathit{K}\left(\mathit{x},\mathit{y}\right)$. For brevity, these sources will be called leaf sources. Their number in a leaf is denoted $N_L$.

The first step of creating clusters consists of establishing the hierarchy of leaf sources. To this purpose, the intensity (scalar or vector) of a source is characterized by its norm. The latter is taken from Euclidian. Thus when the intensity is a scalar, the norm equals its absolute value; for a vector, it is the square root from the sum of squared components. The norm $q=\left|\mathit{q}\right|$ of intensity $\mathit{q}$ gives the source magnitude. Of essence is that a magnitude is a non-negative scalar. This serves to establish the hierarchy of the leaf sources by their assembling in the order of decreasing magnitudes $q_1\ge q_2\ge \dots \ge q_{N_L}\ge 0,q_j=\left|{\mathit{q}}_j\right|,j=1,\dots , N_L$. The ordering is efficiently performed by using a standard procedure, say Quicksort. This path of establishing hierarchy has been effectively applied for zero-order moments [25] to reduce the total number of M2L translations. It is equally applicable to the leaf sources subjected to S2M translations.

The second step of clustering consists of using the established hierarchy for successive grouping of the sources into clusters. The grouping is performed as follows. The source with the highest magnitude becomes the object around which the first cluster is grouped. This is the leading source for the cluster. The grouping may be performed in a standard way, by using distance connectivity. Indeed, with known distances between sources, the metric distance matrix [31], is actually known. Thus we may distinguish as a cluster the leading source itself and all the sources, whose distances to it do not exceed an assigned cluster radius $r_{cl}$ (Figure 1). The latter may be chosen as convenient. As mentioned, for a leaf with the equivalent surface (ES) of radius $R_e$, it is appropriate to take the cluster radius $r_{cl}$ not exceeding $0.125 R_e$. To the moment, the cluster radius $r_{cl}$ is assumed set.

A similar process is applied to the remaining sources in the hierarchical ordering of sources. This defines the second cluster, grouped around the second leading source and separated from the first cluster. The process is continued until all the sources of the leaf are distributed between clusters. Finally, the collection of leaf sources is represented by the collection of clusters. Thus $N_L={\sum }_{p=1}^{N_{cl}}{M}_{clp}$, where $N_{cl}$ is the total number of clusters, $M_{clp}$ is the number of sources in the $p$-th cluster.

The third step consists of appropriately characterizing a cluster. We characterize each of the clusters quantitatively by introducing its representative source. The latter is a specific pointed source with known intensity ${\mathit{q}}_{clp}$ and location ${\mathit{y}}_{clp}$. In cases, when sources depend on the normal at points of their location, the cluster is additionally supplied with known normal ${\mathit{n}}_{clp}$. Specifically, the intensity, location and normal of the representative source of the $p$-th cluster are defined as the sums

$${\mathit{q}}_{clp}={\sum }_{j=1}^{M_{clp}}{\mathit{q}}_j, {\mathit{y}}_{clp}={\displaystyle \frac{1}{\left|{\mathit{q}}_{clp}\right|}}{\sum }_{j=1}^{M_{clp}}\left|{\mathit{q}}_j\right|{\mathit{y}}_j, {\mathit{n}}_{clp}={\displaystyle \frac{{\sum }_{j=1}^{M_{clp}}\left|{\mathit{q}}_j\right|{\mathit{n}}_j}{\left|{\sum }_{j=1}^{M_{clp}}\left|{\mathit{q}}_j\right|{\mathit{n}}_j\right|}} ,\hspace{1em}\hspace{1em}p=1,\dots , N_{cl}$$

(1)

In (1), the summation index $j$presumes an internal enumeration of the sources. The first equation, in contrast with the two others, contains a summation of the vectors of intensities rather than their magnitudes. In the case, when the kernel depends on the normal at a source point and the vector ${\sum }_{j=1}^{M_{clp}}\left|{\mathit{q}}_j\right|{\mathit{n}}_j$ is zero-vector, the sources mutually annihilate; this means that the cluster is of zero intensity. Then it is reasonable to set ${\mathit{q}}_{clp}={\displaystyle 0}$.

These steps are performed only once before starting iterations by up- and downward runs over the hierarchical tree. Merely intensities of sources, comprising the clusters, are updated at an iteration. The clusters, as collections of the sources, are not changed. For further discussion of essence is the following consequence of the first of Equation (1). With growing distance $\left|\mathit{x}-\mathit{y}\right|$ from a cluster, the total field of sources, comprising it, tends toward the field of the representative source. Therefore, the total field from all leaf sources tends toward the sum of fields from representative sources.

3. Enhanced Translations

As mentioned, far fields of leaf sources, beyond the layer of the thickness $0.25R_e$, are translated conventionally much more accurately than far fields of sources within the layer. Thus translations for the sources with distances to the leaf center less than $0.75R_e$ may be translated conventionally. They may be excluded from the clustering process, presented in the previous section.

The discussion focuses on sources comprising the distinguished clusters. For them improving the accuracy of S2M translations appears desirable. From now on, without loss of generality, to simplify notations, we assume that the leaf contains merely the sources comprising the clusters. The total number of these sources is $N_L$, and the number of clusters is $N_{cl}$. Their intensities, locations and normals of the representative sources, are defined by (1) for each of the clusters. Specifically, for the $p$-th cluster, they are ${\mathit{q}}_{clp}$, ${\mathit{y}}_{clp}$, and ${\mathit{n}}_{clp}$ ($p=1,\dots ,N_{cl}$).

Recall how upward translations are performed conventionally by a KI-FMM (e.g., [18,20]). We shall use the interpretation of a translation in terms of equivalent intensities [24]. For a given kernel $\mathit{K}(\mathit{x},\mathit{y})$, the exact far field $\mathit{f}\left(\mathit{x}\right)$, translated to a field point $\mathit{x}$ from leaf sources, is

$$\mathit{f}\left(\mathit{x}\right)={\sum }_{j=1}^{N_L}\mathit{K}(\mathit{x},{\mathit{y}}_j){\mathit{q}}_j$$

(2)

In (2), $N_L$ is the number of sources in the leaf, whose far fields are generated by the kernel $\mathit{K}(\mathit{x},\mathit{y})$, ${\mathit{q}}_j$ are their intensities, ${\mathit{y}}_j$ are points of their locations. When the kernel $\mathit{K}\left(\mathit{x},\mathit{y}\right)$ depends on the normal at the source point ${\mathit{y}=\mathit{y}}_j$, the normal ${\mathit{n}}_j$ assigned at ${\mathit{y}}_j$ is taken into account.

A KI-FMM approximates the exact field (2) by replacing the field of the leaf sources by the field, generated by sources at an equivalent surface (ES):

$$\mathit{f}\left(\mathit{x}\right)\approx {\sum }_{s=1}^{N_e}\mathit{K}(\mathit{x},{\mathit{y}}_{es}){\mathit{q}}_{es}$$

(3)

Herein, $N_e$ is the total number of equivalent sources with intensities ${\mathit{q}}_{es}$ located at nodes ${\mathit{y}}_{es}$ on the ES. In cases when the kernel depends on the normal at the source point, the normal is orthogonal to the ES.

The equivalent intensities ${\mathit{q}}_{es}$ are found by equating the right-hand sides of (3) to the exact field at $N_e$ check points ${\mathit{x}}_i$ located at a check surface (ChS):

$${\sum }_{s=1}^{N_e}\mathit{K}\left({\mathit{x}}_i,{\mathit{y}}_{es}\right){\mathit{q}}_{es}={\sum }_{j=1}^{N_L}\mathit{K}\left({\mathit{x}}_i,{\mathit{y}}_j\right){\mathit{q}}_{j },\hspace{1em}\hspace{1em}i=1,\dots ,N_e$$

(4)

For certainty, the ES and ChS may be assumed spherical of radii $R_e$ and $R_{ch}$, respectively. The nodes are distributed uniformly; the normal at a node at ChS, like the normal at a point at ES, is orthogonal to the surface. Such choices minimize the translation error for a given number $N_e$, as compared with other choices, say, with cubic ES [24]. The system (4) is solved by using its inverse matrix. The latter is calculated in advance and stays the same in all iterations. The found equivalent intensities ${\mathit{q}}_{es}$ are used in Equation (3), which approximates the far field (2) at any point outside the ChS. This completes the path of the conventional method of translation.

The modified translations employ the clusters and their representative sources to increase the accuracy of translated far fields. As mentioned, the clustering algorithm includes all $N_L$ leaf sources, so that $N_L={\sum }_{p=1}^{N_{cl}}{M}_{clp}$. Hence the far field (2) may be evaluated as the sum of far fields of clusters

$$\mathit{f}\left(\mathit{x}\right)={\sum }_{p=1}^{N_{cl}}{\mathit{f}}_p\left(\mathit{x}\right)$$

where ${\mathit{f}}_p\left(\mathit{x}\right)$ is the far field of the cluster $p$. It is

$${\mathit{f}}_p\left(\mathit{x}\right)={\sum }_{j=1}^{M_{clp}}\mathit{K}(\mathit{x},{\mathit{y}}_j){\mathit{q}}_j$$

(5)

Consider a particular cluster $p$. Its representative source has the intensity ${\mathit{q}}_{clp}$, location ${\mathit{y}}_{clp}$, and normal ${\mathit{n}}_{clp}$, defined by (1). As emphasized in the previous section, with growing distance $\left|\mathit{x}-\mathit{y}\right|$ from a cluster to a field point $\mathit{x}$, the field ${\mathit{f}}_p\left(\mathit{x}\right)$ tends to the field of its representative source. Thus the difference between the fields of the cluster and its representative source goes to zero. Clearly, it decreases faster than the field of the cluster. Of essence is also that the far field of the representative source is exactly evaluated as the product $\mathit{K}(\mathit{x},{\mathit{y}}_{clp}){\mathit{q}}_{clp}$.

This suggests using the “subtract-and-add” approach by subtracting from and adding to the right-hand side of (5) the known field $\mathit{K}(\mathit{x},{\mathit{y}}_{clp}){\mathit{q}}_{clp}$ of the representative source. Then (5) is written as

$${\mathit{f}}_p\left(\mathit{x}\right)={\mathit{f}}_{mp}\left(\mathit{x}\right)+\mathit{K}\left(\mathit{x},{\mathit{y}}_{clp}\right){\mathit{q}}_{clp}$$

where the field ${\mathit{f}}_{mp}\left(\mathit{x}\right)$ is the difference of the fields:

$${\mathit{f}}_{mp}\left(\mathit{x}\right)={\mathit{f}}_p\left(\mathit{x}\right)-\mathit{K}(\mathit{x},{\mathit{y}}_{clp}){\mathit{q}}_{clp}$$

(6)

The difference will be called the residual (modified) field. As mentioned, the modified field decreases faster than the field of the cluster itself. Consequently, it is reasonable to apply the approximation of (3) type to the residual field, rather than to the field of the cluster. Thus, we approximate the modified field similar to (3)

$${\mathit{f}}_{mp}\left(\mathit{x}\right)\approx {\sum }_{s=1}^{N_e}{\mathit{K}}_{mp}\left(\mathit{x},{\mathit{y}}_{es}\right){\mathit{q}}_{mesp}$$

(7)

where ${\mathit{q}}_{mesp}$ are modified equivalent intensities of sources located on the ES; ${\mathit{K}}_{mp}\left(\mathit{x},\mathit{y}\right)$ is the modified kernel. It is defined as the difference

$${\mathit{K}}_{mp}\left(\mathit{x},\mathit{y}\right)=\mathit{K}\left(\mathit{x},\mathit{y}\right)-\mathit{K}(\mathit{x},{\mathit{y}}_{clp})$$

(8)

Definition (8) guarantees that with growing distance $\left|\mathit{x}-\mathit{y}\right|$, the right-hand side of (7) goes to the exact modified field ${\mathit{f}}_{pm}\left(\mathit{x}\right)$.

Similar to ${\mathit{q}}_{es}$ in (3), the modified equivalent intensities ${\mathit{q}}_{mesp}$ of cluster $p$ are found by equating the right-hand side of (7) to the exact modified field (6) at $N_e$ check points ${\mathit{x}}_i$ at the ChS

$${\sum }_{s=1}^{N_e}{\mathit{K}}_{mp}\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{es}\right){\mathit{q}}_{mesp}={\sum }_{j=1}^{M_{clp}}\mathit{K}\left({\mathit{x}}_i,{\mathit{y}}_j\right){\mathit{q}}_j-\mathit{K}\left({\mathit{x}}_i,{\mathit{y}}_{clp}\right){\mathit{q}}_{clp},\hspace{1em}\hspace{1em}i=1,\dots ,N_e$$

(9)

The ES and ChS are taken the same as those used in (3) and (4) for conventional translations. The system (9) is solved by using its inversed matrix. As for the conventional matrix in (4), the latter is calculated in advance and stays the same in all iterations.

The found modified equivalent intensities ${\mathit{q}}_{mesp}$ are used in Equation (7), which approximates the modified far field ${\mathit{f}}_{mp}\left(\mathit{x}\right)$ defined by (6). Then the sum ${\mathit{f}}_p\left(\mathit{x}\right)={\mathit{f}}_{mp}\left(\mathit{x}\right)+\mathit{K}(\mathit{x},{\mathit{y}}_{clp}){\mathit{q}}_{clp}$ gives the far field of the cluster.

In this way, far fields ${\mathit{f}}_p\left(\mathit{x}\right)$ of all the clusters $(p=1,\dots ,N_{cl})$ become known. Finally, their sum $\mathit{f}\left(\mathit{x}\right)\approx {\sum }_{p=1}^{N_{cl}}{\mathit{f}}_p\left(\mathit{x}\right)$ gives the needed approximation of the far field (2) of all leaf sources. This completes the modified translation suggested. The total algorithm of the modified translations, including its clustering stage described in the previous section, is summarized in Appendix A. Emphasize, that the first two steps of the algorithm are performed only once and before starting iterations. As mentioned, merely intensities of sources, comprising the clusters, are updated at an iteration. Meanwhile, the two remaining steps, involving modified matrices, do not differ in complexity from the conventional method. Consequently, the time expense is actually the same for the modified and conventional approaches.

We find the far field (2) approximately in the way, different from that used in conventional S2M translations. This path provides the gain in accuracy due to exact evaluation of the far fields of the representative sources, whose input into the field $\mathit{f}\left(\mathit{x}\right)$ is highest. This is clearly demonstrated by examples in Section 5. The additional gain is due to favorable for the accuracy fast decreasing of the modified (residual) fields.

Emphasize that if a cluster, say for certainty the first ($p=1$), contains only one (leading) source, then $M_{cl1}=1$ and the leading source becomes the representative source. In this case ${\mathit{y}}_1={\mathit{y}}_{cl1}$, ${\mathit{q}}_1={\mathit{q}}_{cl1}$. Hence, the right-hand side of (9) is zero. For kernels used in practice, the equations arising for different check nodes are linearly independent. Therefore, the determinant of the system (9) is non-zero; consequently, for zero right-hand side, the solution, giving the modified equivalent intensities, is zero, as well: ${\mathit{q}}_{mes1}={\displaystyle 0}$ ($s=1,\dots ,N_e$). Then approximation of ${\mathit{f}}_{m1}\left(\mathit{x}\right)$ by the right-hand side of (7) is zero. This is the exact modified field ${\mathit{f}}_{m1}\left(\mathit{x}\right)$ defined by (6), because in the case considered, ${\mathit{f}}_{m1}\left(\mathit{x}\right)=\mathit{K}(\mathit{x},{\mathit{y}}_1){\mathit{q}}_1-\mathit{K}(\mathit{x},{\mathit{y}}_{cl1}){\mathit{q}}_{cl1}\equiv {\displaystyle 0}$.

We see, that when the cluster radius $r_{cl}$ or/and intensities of each of its sources, except for the leading one, are very small (in the limit, go to zero), and the error of the modified translation becomes negligibly small. In a general case, it is less than the error of the conventional S2M translation applied to sources of the cluster.

The modified translations have been suggested above by using quite general considerations regarding “fast decreasing” of a field, “gain in accuracy”, etc. Explaining these concepts in quantitative terms may be performed theoretically or/and numerically. Theoretical study requires special investigations, which would be especially involved for 3D Navier kernels (singular and hypersingular). In this paper, we prefer to avoid rigorous mathematical analyses and to obtain useful for practical applications estimations numerically. Specifically, the gain in accuracy is provided by direct calculations and comparison of the errors of the conventional and modified methods. The maximal and average errors are found for the kernels of direct BIE of 2D and 3D Laplace and Navier equations, solved by using the KI-FMM.

4. Numerical Results for Typical Kernels of BIE

4.1. Computational Prerequisites

Consider kernels entering the singular and hypersingular BIE of the Laplace and Navier equations. To keep track of the results of the paper [24], where the accuracy was studied for conventional translations, we shall use the notations and parameters of the cited paper. Thus, in this section, when specifying a kernel, we use normal (not bold) typescript for matrices and vectors.

The starting singular and hypersingular BIE are

$${\int }_{S}W\left(x,y\right){\mathsf{\Delta }q}_{ny}\left(y\right)d{S}_y+{\int }_S{G}_{ny}\left(x,y\right)\mathsf{\Delta }u\left(y\right)d{S}_y=0.5u\left(x\right),\hspace{1em}\hspace{1em}x\in S$$

(10)

$${\int }_S{Q}_{nx}\left(x,y\right){\mathsf{\Delta }q}_{ny}\left(y\right)d{S}_y+{\int }_S{H}_{nx,ny}\left(x,y\right)\mathsf{\Delta }u\left(\mathrm{y}\right)d{S}_y=0.5q_{nx}\left(x\right),\hspace{1em}\hspace{1em}x\in S$$

(11)

Herein, $S$ is a surface (open or closed), on which the potential or/and the flux may be discontinuous. The surface is assumed smooth enough to have the classical factor 0.5 on the right-hand side. $W\left(x,y\right)$ is a weakly singular kernel. $G_{ny}\left(x,y\right)$ is a field of double-layer potential; the subscript $ny$ indicates that the kernel depends on the normal to the surface $S$ at an integration point $y$; the dependence on the normal is of prime significance for the further analysis of the accuracy. $Q_{nx}\left(x,y\right)$ is the singular kernel; similar to $W\left(x,y\right)$, it does not depend on the normal at the integration point. $H_{nx,ny}\left(x,y\right)$ is the hypersingular kernel; similar to $G_{ny}\left(x,y\right)$, it depends on this normal; $q_{ny}$ is the normal component of the flux $q$ on $S$. ${\mathsf{\Delta }q}_{ny}=q_{ny}^{+}-q_{ny}^{-}$ is the discontinuity of the flux at the integration point $y$; $\mathsf{\Delta }u=u^{+}-u^{-}$ is the discontinuity of the potential $u$. The sign “plus” (“minus”) marks the limiting values from the side with respect to which the normal is outward (inward). For external boundaries, the normal is assumed outward. Then, for a given potential $u$ (flux component $q_{ny}$) on it, we set $u^{+}=u$, $u^{-}=0$ ($q_{ny}^{+}=q_{ny}$, $q_{ny}^{-}=0$). For the Laplace equation, the potential $u$ is scalar; for the Navier equation, it is a vector.

Similar to [24], in the calculations discussed below, in 2D, the leaf cell is a square with half-sides of unit length. The equivalent surface (ES) is a circle of radius $R_e=1.415$. The radii of the check (ChS) and control surfaces are the same $R_{Ch}=R_{Cont}=2.98$. The nodes at all surfaces are located uniformly; their numbers are $N_{nd}=24$ for the ES and ChS; and $N_{Cont}=$ 1000 for the control surface.

In 3D, the leaf cell is a cube with sides of unit length; the ES is spherical with a radius $R_e=0.866$. For the Laplace equation, the radii of the check (ChS) and control surfaces are the same $R_{Ch}=R_{Cont}=1.49$. The nodes on them are located almost uniformly; their numbers are $N_{nd}=218$ for the ES and ChS. The nodes at the control surface are taken in the cross-section coinciding with the plane of axes $x_1$ and $x_2$; their number in this cross-section is $N_{Cont}=1580$. For the Laplace equation, the translated field is a scalar, which becomes zero at some points of the control surface. Thus the maximal absolute errors are studied.

For the Navier equation, the number of translated components is three-fold greater than for the 3D Laplace BIE. To maintain the accuracy, larger number of nodes $N_{nd}=386$ is used for the ES and ChS. The radius of the ChS is $R_{Ch}=1.49$. That of the control surface is $R_{Cont}=2.00$. The control points are uniformly located at each of the 18 principle circles. The normal to a circle plane is taken orthogonal to the $x_2$-axis, its angle with the $x_1$-axis is changed with the step $10°$. Along a principle circle, there are 72 control points. Thus the total number of control points is $18·72=1296$. For the Navier equation, the translated quantities are vectors which in the examples considered are non-zero. Thus for them, the maximal relative errors are studied using the Euclidian norm of the vectors.

We consider a cluster comprised of sources with unit intensity each. Using equal intensities is the most unfavourable for the modified translation, because there is no leading source with intensity much greater than that of the remaining sources. The number of sources, comprising a cluster, is 30 in 2D and 50 in 3D. Their centres are distributed almost uniformly by using the sunflower spiral (e.g., [32]). For generality, the radius $r_{cl}$ of the cluster is defined by the ratio $r_{cl}/R_e$, where the radius $R_e$ of the ES is the same as taken above for leaf cells in 2D and 3D. The ratio $r_{cl}/R_e$ is changed from 0.01 (sometimes from 0.07) to the maximal value 0.12 for which the cluster diameter ${2r}_{cl}$ approximately equals to the thickness $h=0.25R_e$ of the layer (Figure 1), where errors of conventional translation, as established in [24], are greatest. The cluster center $y_{clp}$ is located at the $x_1$ axis, so that its coordinates are ${(x}_1$, 0) in 2D and ${(x}_1$, 0, 0) in 3D with $x_1\ge 0$. The normal to the ES at its point closest to the cluster center is $(1, 0)$ in 2D and $(1, 0, 0)$ in 3D. For kernels $G_{ny}\left(x,y\right)$ and $H_{nx,ny}\left(x,y\right)$, depending on the normal $n_y$ at a source point, the vector $n_y$ is taken in the most unfavourable direction which is orthogonal to the normal at the ES. Thus for these kernels, $n_y=(0, 1)$ in 2D, and $n_y=(0, 1, 0)$ in 3D; the same direction is taken for each source comprising a cluster.

The distance $R_e-\left|y_{clp}\right|$ from the cluster to the ES is $R_e-x_1$. Again it is defined by the ratio $\epsilon =(R_e-\left|y_{clp}\right|)/R_e$. To see how the errors of conventional and modified translations decrease with the distance, the calculations were performed for various ratios, which changed from 0.02 (sometimes from 0.1) to 0.2.

The maximal and average errors of conventional and modified translations are found by the comparison of exact and translated far fields of the cluster at each point of the control surface. The exact far field is calculated in accordance with (2) by summing fields of sources comprising the cluster. The far field, corresponding to the conventional translations, is calculated by using Equations (3) and (4). The far field, corresponding to the modified translations, is calculated by means of (6)–(9) by employing the algorithm summarised in Appendix A.

4.2. 2D Laplace BIE

Translation by means of weakly singular kernel $W\left(x,y\right)$. In this case, the kernel $K(x,y)$ in the conventional (4) and modified (9) translation is $W\left(x,y\right)=-{\displaystyle \frac{1}{2\pi k}}\mathrm{l}\mathrm{n}\left|r\right|$, where, $k$ is the conductivity assumed unit ($k=1$); $r=\left|x-y\right|$.

Consider the quite unfavorable case, when the representative source is located at the relative distance $\epsilon =(R_e-\left|y_{clp}\right|)/R_e=0.2$ from the ES and the relative radius of the cluster is $r_{cl}/R_e=0.12$. Figure 2 presents the comparative distributions of the absolute error along the angular coordinate (in radians) of the control surface. The maximal errors occur near the middle of an interval between two neighbouring control points. The minimal error is zero, as it must be when a control point coincides with a check point.

For various distances $\epsilon$ from the ES and for various sizes $r_{cl}$ of the cluster, the maximal and average errors are summarized in

Table 1. Maximal and average errors for translations by means of weakly singular kernel (2D singular Laplace BIE).

	Conventional			Modified
$\mathit{\epsilon }$	0.1	0.2		0.1	0.2
${\mathit{r}}_{\mathit{c}\mathit{l}}/{\mathit{R}}_{\mathit{e}}$	0.07	0.07	0.12	0.07	0.07	0.12
max	5.78 × 10−6	3.13 × 10−6	3.50 × 10−6	8.32 × 10−7	2.60 × 10−7	9.06 × 10−7
average	1.20 × 10−6	6.37 × 10−7	6.36 × 10−7	1.43 × 10−7	4.57 × 10−8	1.59 × 10−7

. It may be seen, that the accuracy of both conventional and modified translations decreases with decreasing distance $\epsilon$ and with growing radius $r_{cl}$.

Both Figure 2 and Table 1 evidently demonstrate a significant increase in the accuracy when using the modified translations as compared with the conventional approach. In particular, for sources located at the distance $\epsilon$ < $0.25R_e$, even for a cluster with a large relative radius $r_{cl}/R_e=0.12$, the accuracy of the modified translations is at least three-fold better than that of the conventional translations.

Translation by means of singular kernel $G_{ny}\left(x,y\right)$. The kernel is $G_{ny}\left(x,y\right)=-{\displaystyle \frac{1}{2\pi }}{\displaystyle \frac{1}{r^2}}{r}_s{n}_{ys}$ (Einstein summation convention is presumed). The kernel depends on the direction of the normal at a source point.

The comparative distributions of errors along the control surface are provided in Figure 3. The maximal and average errors for various distances $\epsilon$ and radii $r_{cl}$ of the cluster are summarized in

Table 2. Maximal and average errors for translations by means of singular kernel (2D singular Laplace BIE).

	Conventional			Modified
$\mathit{\epsilon }$	0.1	0.2		0.1	0.2
${\mathit{r}}_{\mathit{c}\mathit{l}}/{\mathit{R}}_{\mathit{e}}$	0.07	0.07	0.12	0.07	0.07	0.12
max	0.00058	0.00017	0.00017	6.36 × 10−6	8.53 × 10−7	3.26 × 10−6
average	8.82 × 10−5	2.42 × 10−5	2.44 × 10−5	9.19 × 10−7	1.35 × 10−7	5.08 × 10−7

.

Again, the results clearly show that the modified translations provide notable gains in accuracy.

Translation by means of the hypersingular kernel $H_{nx,ny}\left(x,y\right)$. The kernel is $H_{nx,ny}\left(x,y\right)=-{\displaystyle \frac{k}{2\pi }}{\displaystyle \frac{1}{r^2}}\left({n_{xj}n}_{yj}-{\displaystyle \frac{2}{r^2}}{r}_j{n_{xj}r}_s{n}_{ys}\right)$. The kernel depends on the direction of the normal at a source point.

The comparative maximal and average errors are summarized in

Table 3. Maximal and average errors for translations by means of hypersingular kernel (2D hypersingular Laplace BIE).

	Conventional			Modified
$\mathit{\epsilon }$	0.1	0.2		0.1	0.2
${\mathit{r}}_{\mathit{c}\mathit{l}}/{\mathit{R}}_{\mathit{e}}$	0.07	0.07	0.12	0.07	0.07	0.12
max	0.00269	0.00082	0.00083	2.87 × 10−5	3.88 × 10−6	1.47 × 10−5
average	0.00056	0.00015	0.00015	5.82 × 10−6	8.54 × 10−7	3.21 × 10−6

.

Again, there is an order gain in the accuracy when using the modified translation. It reaches two orders for the distance $\epsilon$ = 0.1 and radius $r_{cl}=0.07R_e$. Further decreasing the distance to $\epsilon =0.002$ and radius to $r_{cl}/R_e=0.001$ provides additionally five orders gain in the accuracy of the modified method. Therefore, it may be expected that for a small radius $r_{cl}$, the cluster of sources may be replaced with a single (representative) source. This option is discussed in the next section.

4.3. 3D Laplace BIE

Consider the kernels entering the hypersingular BIE (11), whose kernels are quite unfavourable for the accuracy of S2M translations.

Translation by means of singular kernel $Q_{nx}\left(x,y\right)$. The kernel is $Q_{nx}\left(x,y\right)=-{\displaystyle \frac{1}{4\pi }}{\displaystyle \frac{1}{{\mathit{r}}^3}}{r}_j{n}_{xj}$.

Table 4. Maximal and average errors for translations by means of singular kernel (3D hypersingular Laplace BIE).

	Conventional				Modified
$\mathit{\epsilon }$	0.02	0.1	0.2		0.02	0.1	0.2
${\mathit{r}}_{\mathit{c}\mathit{l}}/{\mathit{R}}_{\mathit{e}}$	0.01	0.07	0.07	0.12	0.01	0.07	0.07	0.12
max	2.39 × 10−5	1.11 × 10−5	7.55 × 10−6	7.52 × 10−6	1.31 × 10−8	5.88 × 10−8	2.22 × 10−8	7.98 × 10−8
average	4.75 × 10−6	2.24 × 10−6	2.01 × 10−6	2.01 × 10−6	3.15 × 10−9	1.80 × 10−8	5.63 × 10−9	1.79 × 10−8

presents the maximal and average errors of the conventional and modified approaches.

It may be seen, that the accuracy of the modified translations is about two orders higher than that of the conventional approach.

Translation by means of hypersingular kernel $H_{nx,ny}\left(x,y\right)$. The kernel is $H_{nx,ny}\left(x,y\right)=-{\displaystyle \frac{k}{4\pi }}{\displaystyle \frac{1}{{\mathit{r}}^3}}\left({n_{xj}n}_{yj}-{\displaystyle \frac{3}{{\mathit{r}}^2}}{r}_j{n_{xj}r}_s{n}_{ys}\right)$. The kernel depends on the direction of the normal at a source point.

Table 5. Maximal and average errors for translations by means of hypersingular kernel (3D hypersingular Laplace BIE).

	Conventional				Modified
$\mathit{\epsilon }$	0.02	0.1	0.2		0.02	0.1	0.2
${\mathit{r}}_{\mathit{c}\mathit{l}}/{\mathit{R}}_{\mathit{e}}$	0.01	0.07	0.07	0.12	0.01	0.07	0.07	0.12
max	0.00012	0.00010	9.68 × 10−5	9.68 × 10−5	4.35 × 10−8	2.42 × 10−7	8.08 × 10−8	2.29 × 10−7
average	4.94 × 10−5	4.80 × 10−5	4.84 × 10−5	4.85 × 10−5	2.17 × 10−8	1.04 × 10−7	3.65 × 10−8	1.05 × 10−7

presents the maximal and average errors.

As could be expected, for each of the distances $\epsilon$ and radii $r_{cl}$, the modified translations provide at least an order higher accuracy.

4.4. 3D Navier BIE

The calculations above show that, similar to the conclusions of [24], the errors of translation are greater for kernels of hypersingular BIE (11) than those of singular BIE (10). For 3D BIE, they are larger than 2D. Furthermore, the errors are greatest when translations are performed by means of a kernel depending on the direction of the normal at a source point. Thus, for the 3D Navier BIE, we consider the BIE (11) with emphasis on the hypersingular kernel.

Translation by means of singular kernel $Q_{nx,ik}\left(x,y\right)$. The kernel is $Q_{nx;ik}\left(x,y\right)={\displaystyle \frac{1}{8\pi (1-\nu )}}{\displaystyle \frac{1}{r^3}}{n}_{xj}\left[{\left(1-2\nu \right)(\delta }_{ij}{r}_k-r_i{\delta }_{jk}-r_j{\delta }_{ik})-{\displaystyle \frac{3}{r^2}}{r}_i{r}_j{r}_k\right]$, where $\nu$ is the Poisson ratio.

The maximal and average relative errors are provided in

Table 6. Maximal and average relative errors for translations by means singular kernel (3D hypersingular Navier BIE).

	Conventional				Modified
$\mathit{\epsilon }$	0.02	0.1	0.2		0.02	0.1	0.2
${\mathit{r}}_{\mathit{c}\mathit{l}}/{\mathit{R}}_{\mathit{e}}$	0.01	0.07	0.07	0.12	0.01	0.07	0.07	0.12
max	0.02166	0.00809	0.00212	0.00189	8.80 × 10−5	0.00058	0.00023	0.00063
average	0.00312	0.00094	0.00035	0.00029	1.11 × 10−5	9.23 × 10−5	3.99 × 10−5	0.00011

.

Similar to the results for the Laplace equation, in all the cases, the accuracy of the modified translations is better than that of the conventional approach. When the relative radius of the cluster does not exceed 0.07, the gain in the accuracy is an order, at least.

Translation by means of hypersingular kernel $H_{nx,ny}\left(x,y\right)$. The kernel is $H_{nx,ny;ik}\left(x,y\right)={\displaystyle \frac{2\mu }{8\pi (1-\nu )}}{\displaystyle \frac{1}{r^3}}{n}_{xj}\left\{\left(1-4\nu \right){\delta }_{ij}{n}_{yk}-\left(1-2\nu \right)\left(n_{yi}{\delta }_{jk}+n_{yj}{\delta }_{ik}\right)-{\displaystyle \frac{3}{r^2}}\left[r_s{n}_{ys}\left(\left(1-2\nu \right){\delta }_{ij}{r}_k+\right.\right.\right.\left.\nu \left(r_i{\delta }_{jk}+r_j{\delta }_{ik}\right)\right)+\left.\left.\nu \left({n_{yi}r}_j+{n_{yj}r}_i\right)r_k+\left(1-2\nu \right)r_i{r}_j{n}_{yk}\right]+{\displaystyle \frac{15}{r^4}}{r}_s{n}_{ys}{r}_i{r}_j{r}_k\right\}$.

The kernel depends on the direction of the normal at a source point, where $\mu$ is the shear modulus.

This case is of special significance for fracture mechanics and, as appears from the results of [24], the accuracy of translations for it is worst. Even when using 386 nodes at ES, the maximal relative error of conventional S2M translations may reach 70% when a point source is located at a node of ES; only when using 602 nodes it decreases to 2%. Thus it is quite desirable to improve the accuracy.

Table 7. Maximal and average relative errors for translations by means hypersingular kernel (3D hypersingular Navier BIE).

	Conventional				Modified
$\mathit{\epsilon }$	0.02	0.1	0.2		0.02	0.1	0.2
${\mathit{r}}_{\mathit{c}\mathit{l}}/{\mathit{R}}_{\mathit{e}}$	0.01	0.07	0.07	0.12	0.01	0.07	0.07	0.12
max	0.06542	0.03746	0.03044	0.02570	0.00046	0.00647	0.00299	0.00837
average	0.00931	0.00532	0.00338	0.00292	6.24 × 10−5	0.00076	0.00036	0.00101

presents the comparison of relative errors for conventional and modified translations when the far field is generated by a cluster of radius $r_{cl}$. Its center is at the relative distance $\epsilon$ from the ES.

It may be seen that for the most unfavourably located cluster with its center close to the ES ($\epsilon =0.02$), the modified method provides two orders greater accuracy for 386 nodes at ES. The error is incomparably less than the mentioned error of 70% reported in [24] for this number of nodes. Thus the modified approach avoids the need in notable increase the number of nodes on ES to bring the accuracy of starting S2M translations to an acceptable level.

The said above referred to a single cluster. The modified S2M translations may be applied to each of the $N_{cl}$ clusters distinguished. This significantly improves the accuracy of far fields from sources located in the spherical layer, as compared with the far fields of these sources, translated conventionally.

Underline that, as established, when any of the clusters contains only one source, the error of its modified translation is zero. This may serve to simplify the modified translations.

5. Simplified Enhanced Translations

Each of the modified translations, introduced to increase the accuracy of calculated far fields, requires evaluation the matrix of the system (9) and its inversion. Depending on the location of the cluster center, the matrices are different for various clusters. The calculation of the matrices and their inversion for each of the clusters is an additional computational burden. Despite it is performed only once, of value is to avoid it, for the price of partial loss of accuracy.

The notion of the zero error of the modified translation for a cluster consisting of a single (leading) source suggests the wanted simplification. Indeed, as mentioned, for such a cluster, the modified equivalent intensities ${\mathit{q}}_{mesp}$ ($s=1,\dots ,N_e$) are zero. Consequently, by (7) the modified far field is zero. Then (6) implies that the actual far field ${\mathit{f}}_p\left(\mathit{x}\right)$ of such a cluster $p$ is promptly found as

$${\mathit{f}}_p\left(\mathit{x}\right)=\mathit{K}(\mathit{x},{\mathit{y}}_{clp}){\mathit{q}}_{clp}$$

(12)

Thus, for this cluster, there is no need for evaluation and inversion of the matrix of the system (9). Equation (12) expresses the evident fact that the far field is generated by the particular leaf source, which has served as the leading source for the $p$-th cluster. This trivial result implies that in a general case, when the modified (residual) far field (6) of a cluster $p$ is notably less than the field of the source representing it, the total far field (5) of all its sources may be quite accurately calculated by using (12).

It remains to establish exactly when the simplification (12) is applicable. The following rough estimation and numerical data on the maximal relative errors serve to obtain an answer for typical kernels.

In many applied problems, like those of fracture mechanics, a cluster is comprised of a large flaw, surrounded by many small cracks. The far field, generated by the single vector representing the cluster, is actually the same as the sum of far fields of all its entries. Even in very unfavorable cases, the upper bound of relative maximal error is of order $E{=r}_{cl}/R$, where $R$ is the distance from the cluster center. In particular, the estimation refers to a cluster composed of merely two sources of equal intensities located at the maximal distance $2r_{cl}$ between them. In 3D, the minimal distance between the equivalent surface of a leaf and the check surface of its parent is $2.44R_e$, where $R_e$ is the radius of the child equivalent surface. Then for $r_{cl}<0.125 R_e$, the upper bound of the relative error is $E<0.125/2.44\approx 0.051$. For sources of equal intensities, which are more/less evenly distributed around the leading source, the actual maximal error is notably less.

Table 8. Maximal and average errors of the far field when representing the cluster by its representative source.

$\mathit{\epsilon }$	0.1		0.2
${\mathit{r}}_{\mathit{c}\mathit{l}}/{\mathit{R}}_{\mathit{e}}$	0.03	0.07	0.03	0.07	0.12
2D Laplace hypersingular BIE and hypersingular kernel
max	0.00054	0.00104	0.00052	0.00100	0.00161
average	0.00032	0.00065	0.00032	0.00064	0.00109
3D Laplace hypersingular BIE and hypersingular kernel
max	0.00016	0.00033	0.00016	0.00032	0.00066
average	0.00013	0.00026	0.00013	0.00026	0.00054
3D Navier hypersingular BIE and hypersingular kernel
max	0.00201	0.00582	0.00176	0.00505	0.01934
average	0.00053	0.00117	0.00053	0.00116	0.00299

evidently demonstrates this for a number of typical kernels.

The question arises, if the maximal errors of the orders disclosed are acceptable for using the simplified translation (12) in practical calculations by KI-FMM? To answer the question, recall that the starting intensities of leaf sources at the beginning of iterations are found quite roughly. The intensities notably change on successive iterations of solving an algebraic system of a BEM. The exact solution of this system already contains errors of discretization the boundaries and approximation of the density. Thus, the final maximal error of tenths of a percent is normally acceptable for practice. When solving the same system by a KI-FMM, it is desirable to retain the overall error below this level. The need for notably less maximal error $E$ for an individual translation arises merely due to the accumulation of individual errors. The error grows on a chain of translations from a leaf source to a target source. For a chain consisting of $N_{tr}$ translations, the upper bound of the accumulated overall error increases nearly proportional to $N_{tr} ($[18], p. 610). The number $N_{tr}$ of individual translations in a chain is defined primarily by the number of M2L translations. The latter, as known (e.g., [33]), in 3D problems is at most 189. Thus, the overall error arising after chain translations may be two orders greater than the error $E$ of an individual translation. When the errors of individual translations are on the needed small level, the error of a selected single translation may be $(N_{tr}-1)$-fold greater. In particular, this refers to the singled-out starting S2M translation, which is the first in the chain. This implies that the maximal relative error caused by the simplified improved translation (12), is commonly acceptable for practical calculations. In many cases, even the error on the overestimated level of $E=0.051$ may be acceptable. Furthermore, for the 3D Navier BIE, which is of special significance for fracture mechanics, the accuracy of the simplified approach is on the level of non-simplified modified translations. It is notably better than the accuracy of the conventional translations (cf. Table 7 and Table 8).

We conclude that for the most practical applications, the simplified modified translations (12) provide acceptable accuracy for far fields translated from the clusters distinguished. This excludes the need to build and inversion the matrices for any of them. When doubting, the applicability of the simplification may be checked in preparatory control calculations performed with solving the system (9).

6. Summary

The paper is concerned with using BEM for accurate evaluation of local fields arising near concentrators in structures composed of multiple elements interacting on their contacts. It presents theoretical rationale and a method to increase the accuracy of matrix-to-vector multiplications, performed on iterations of KI-FMM when solving the BEM system.

The method improves the accuracy of fields translated from those sources of the BEM system, whose far fields, when translated conventionally, have the greatest errors. These leaf sources are assembled into clusters by choosing an appropriate leading source, and by grouping around it the sources, located closer than a given threshold. The latter is found in numerical experiments performed for typical kernels. A composed cluster is characterized by the representative source. Its intensity, location and normal (for kernels depending on the normal at the source point) are uniquely defined by the leaf sources comprising it.

For a cluster distinguished, the accuracy of the conventional approximation of its far field is increased by the modified S2M translation suggested. The improved translation is based on the approach “subtract and add”. The approach consists of subtracting and adding the directly calculated input from the representative source, which generates the dominant part of the far field. We define the residual far field as the difference between the true far field from the cluster sources and the far field of the representative source. It decreases much faster than the far field from the sources. This yields that the maximal error of its translation is notably less than that of the conventional approximation of the far field from the sources themselves. The approximation of the residual far field is performed by employing the modified kernel adjusted to its rapid decrease with growing distance. The modified equivalent intensities are found from the algebraic system obtained by equating the approximated residual field to its exact values at check nodes. As usual for the “subtract and add” approach, the gain is due to the exact evaluation of the added principal term. Numerical results for typical kernels of BIE solved by BEM demonstrate the notable increase in accuracy when employing the modified S2M translations.

Composing the clusters, building and inversion of the modified matrices are performed only once before starting iterations. The complexity of modified translations, performed on iterations, is the same as that of conventional translations. As a result, the generation of clusters and using modified translations for improving the accuracy of S2M translations does not increase the time expense.

Furthermore, to the accuracy acceptable for many practical applications, the residual far field may be neglected as compared with the dominant far field of the representative source. This excludes the need to solve the modified system and, consequently, in the evaluation and inversion of the matrices for each of the clusters. Thus, the modified translations are actually reduced to calculation far fields from the representative sources at the nodes on the parent check surface. This additionally notably simplifies the modified S2M translations.

The same simplification is reasonable to use when the number $N_L$ of sources in a leaf is less, close or insignificantly greater than the number $N_e$ of nodes on the ES of the leaf. In these practically important cases, each of the sources may be considered as a cluster, comprised of a single source. Then there is no sense in finding $N_e$ equivalent intensities. The needed values of the total far field are exactly calculated as the sums of fields arising from the $N_L$ sources. Then the time expense is less (when $N_L<N_e$) or on the level (when $N_L~N_e$) of that for calculations involving equivalent intensities.

The method suggested is also applicable to improving the accuracy of M2M upward translations. They translate far fields from children to their parents. For each parent, the collection of sources, located at equivalent surfaces of children, is known. Therefore, the algorithm, summarized in Appendix A, may be applied to this collection.

The extension of the modified method, discussed above in terms of the KI-FMM, to analytical forms of the FMM is immediate. It is sufficient to take into account that, in essence, a leaf source is a node of a boundary element. Its intensity is exactly defined by the zero-order moment. Thus, the S2M translation (conventional or modified) is now performed by means of the truncated analytical expansion of a kernel.