Strong form mesh-free hp-adaptive solution of linear elasticity problem

Abstract

We present an algorithm for hp-adaptive collocation-based mesh-free numerical analysis of partial differential equations. Our solution procedure follows a well-established iterative solve–estimate–mark–refine paradigm. The solve phase relies on the Radial Basis Function-generated Finite Differences (RBF-FD) using point clouds generated by advancing front node positioning algorithm that supports variable node density. In the estimate phase, we introduce an Implicit-Explicit (IMEX) error indicator, which assumes that the error relates to the difference between the implicitly obtained solution (from the solve phase) and a local explicit re-evaluation of the PDE at hand using a higher order approximation. Based on the IMEX error indicator, the modified Texas Three Step marking strategy is used to mark the computational nodes for h-, p- or hp-(de-)refinement. Finally, in the refine phase, nodes are repositioned and the order of the method is locally redefined using the variable order of the augmenting monomials according to the instructions from the mark phase. The performance of the introduced hp-adaptive method is first investigated on a two-dimensional Peak problem and further applied to two- and three-dimensional contact problems. We show that the proposed IMEX error indicator adequately captures the global behaviour of the error in all cases considered and that the proposed hp-adaptive solution procedure significantly outperforms the non-adaptive approach. The proposed hp-adaptive method stands for another important step towards a fully autonomous numerical method capable of solving complex problems in realistic geometries without the need for user intervention.

Full text

Vol.:(0123456789)
1 3
Engineering with Computers (2024) 40:1027–1047
https://doi.org/10.1007/s00366-023-01843-6
ORIGINAL ARTICLE
Strong form mesh‑free hp‑adaptive solution oflinear elasticity
problem
MitjaJančič1,2 · GregorKosec1
Received: 25 October 2022 / Accepted: 4 May 2023 / Published online: 27 May 2023
© The Author(s) 2023
Abstract
We present an algorithm for hp-adaptive collocation-based mesh-free numerical analysis of partial differential equations.
Our solution procedure follows a well-established iterative solve–estimate–mark–refine paradigm. The solve phase relies
on the Radial Basis Function-generated Finite Differences (RBF-FD) using point clouds generated by advancing front node
positioning algorithm that supports variable node density. In the estimate phase, we introduce an Implicit-Explicit (IMEX)
error indicator, which assumes that the error relates to the difference between the implicitly obtained solution (from the
solve phase) and a local explicit re-evaluation of the PDE at hand using a higher order approximation. Based on the IMEX
error indicator, the modified Texas Three Step marking strategy is used to mark the computational nodes for h-, p- or hp-
(de-)refinement. Finally, in the refine phase, nodes are repositioned and the order of the method is locally redefined using
the variable order of the augmenting monomials according to the instructions from the mark phase. The performance of
the introduced hp-adaptive method is first investigated on a two-dimensional Peak problem and further applied to two- and
three-dimensional contact problems. We show that the proposed IMEX error indicator adequately captures the global behav-
iour of the error in all cases considered and that the proposed hp-adaptive solution procedure significantly outperforms the
non-adaptive approach. The proposed hp-adaptive method stands for another important step towards a fully autonomous
numerical method capable of solving complex problems in realistic geometries without the need for user intervention.
Keywords RBF-FD· hp-adaptivity· Mesh-free· Linear elasticity· Error indicator
1 Introduction
Many natural and technological phenomena are modelled
through Partial Differential Equations (PDEs), which can
rarely be solved analytically—either because of geometric
complexity or because of the complexity of the model at
hand. Instead, realistic simulations are performed numeri-
cally. There are well-developed numerical methods that
can be implemented in a more or less effective numerical
solution procedure and executed on modern computers to
perform virtual experiments or simulate the evolution of
various natural or technological phenomena. Nonetheless,
despite the immense computing power at our disposal, which
allows us to solve ever more complex problems numerically,
the development of efficient numerical approaches is still
crucial. Relying solely on brute force computing often leads
to unnecessarily long computations—not to mention wasted
energy.
Most numerical solutions are obtained using mesh-based
methods such as the Finite Volume Method (FVM), the
Finite Difference Method (FDM), the Boundary Element
Method (BEM) or the Finite Element Method (FEM). Mod-
ern numerical analysis is dominated by FEM [1] as it offers a
mature and versatile solution approach that includes all types
of adaptive solution procedures [2] and well understood
error indicators [3]. Despite the widespread acceptance of
FEM, the meshing of realistic 3D domains, a crucial part of
FEM analysis where nodes are structured into polyhedrons
covering the entire domain of interest, is still a problem that
often requires user assistance or development of domain-
specific algorithms [4].
* Mitja Jančič
mitja.jancic@ijs.si
Gregor Kosec
gregor.kosec@ijs.si
1 Parallel andDistributed Systems Laboratory, Institute Jožef
Stefan, Jamova Cesta 39, Ljubljana1000, Slovenia
2 Jožef Stefan International Postgraduate School, Jamova Cesta
39, Ljubljana1000, Slovenia
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

1028 Engineering with Computers (2024) 40:1027–1047
1 3
In response to the tedious meshing of realistic 3D domains,
required by FEM, and the geometric limitations of FDM and
FVM, a new class of mesh-free methods [5] emerged in the
1970s. Mesh-free methods do not require a topological rela-
tionship between computational nodes and can therefore oper-
ate on scattered nodes, which greatly simplifies the discreti-
sation of the domain [6], regardless of its dimensionality or
shape [7, 8]. Just recently, they have also been promoted to
Computer Aided Design (CAD) geometry aware numerical
analysis [9]. Moreover, the formulation of mesh-free methods
is extremely convenient for implementing h-refinement [10],
considering different approximations of partial differential
operators in terms of the shape and size of the stencil [11, 12]
and the local approximation order [13]. However, they tend to
be more computationally intensive as they require larger sten-
cils for stable computations [13, 14] and have limited preproc-
essing capabilities [15]. This may make them less attractive
from a computational point of view, but the ability to work
with scattered nodes and easily control the approximation
order makes them good candidates for many applications in
science and industry [16, 17].
Adaptive solution procedures are essential in problems
where the accuracy of the numerical solution varies spatially
and are currently subject of intensive studies. Two concep-
tually different adaptive approaches have been proposed,
namely p-adaptivity or h-, r-adaptivity. In p-adaptivity, the
accuracy of the numerical solution is varied by changing
the order of approximation, while in h- and r-adaptivity,
the resolution of the spatial discretisation is adjusted for the
same purpose. In the h-adaptive approach, nodes are added
or removed from the domain as needed, while in the r-adap-
tive approach the total number of nodes remains constant
– the nodes are only repositioned with respect to the desired
accuracy. Ultimately, h- and p-adaptivities can be com-
bined to form the so-called hp-adaptivity [18–20], where
the accuracy of the solution is controlled with the order of
the method and the resolution of the spatial discretisation.
Since the regions where higher accuracy is required are often
not known a priori, and to eliminate the need for human inter-
vention in the solution procedure, a measure of the quality of the
numerical solution, commonly called a posterior error indicator,
is a necessary additional step in an adaptive solution procedure
[4]. The most famous error indicator, commonly referred to as
the ZZ-type error indicator, was introduced in 1987 by Zienkie-
wicz and Zhu [21] in the context of FEM and it is still an active
research topic [22]. The ZZ-type error indicator assumes that
the error of the numerical solution is related to the difference
between the numerical solution and a locally recovered solu-
tion. The ZZ-type error indicator has also been employed in
the context of mesh-free solutions of elasticity problems using
the mesh-free Finite Volume Method [23] in both weak and
strong form using the Finite Point Method [24]. Furthermore,
it also served as an inspiration in the context of Radial Basis
Function-Generated Finite Difference (RBF-FD) solution to
Laplace equation [25]. Moreover, a residual-based class of error
indicators [26] has been demonstrated in the elasticity problems
using a Discrete Least Squares mesh-free method [27]. Nev-
ertheless, the most intuitive error indicators are based on the
physical interpretation of the solution, usually evaluating the first
derivative of the field under consideration [11] or calculating
the variance of the field values within the support domain [10].
The advent of hp-adaptive numerical analysis began with
FEM in the 1980s [28]. In hp-FEM, for example, one has the
option of splitting an element into a set of smaller elements
or increasing its approximation order. This decision is often
considered to be the main difficulty in implementing the
hp-adaptive solution procedure and was already studied by
Babuška [28] in 1986. Since then, various decision-making
strategies, commonly referred to as marking strategies, have
been proposed [2, 29]. The early works use a simple Texas
Three Step algorithm, originally proposed in the context of
BEM [30], where the refinement is based on the maximum
value of the error indicator. The first true hp-strategy was
presented by Ainsworth [31] in 1997, since then many others
have been proposed [2, 29]. In general, p- in FEM is more
efficient when the solution is smooth. Based on this observa-
tion, most authors nowadays use the local Sobolev regular-
ity estimate to choose between the h- and the p-refinement
[32–34] for a given finite element. Moreover, in [35] local
boundary values are solved, while the authors of [36, 37]
use minimisation of the global interpolation error methods.
For mesh-free methods, h-adaptivity comes naturally with
the ability to work with scattered nodes, and as such has been
thoroughly studied in the context of several mesh-free meth-
ods [38–40]. Only recently, the popular Radial Basis Func-
tion-generated Finite Differences (RBF-FD) [41] have been
used in the h-adaptive solution of elliptic problems [25, 42]
and linear elasticity problems [10, 43]. Researchers have also
reported the combination of h- and r-adaptivity, which form a
so-called hr-adaptive solution procedure [44]. The p-adaptive
method, on the other hand, is still quite unexplored in the
mesh-free community. However, the authors of [45] approach
the p-adaptive RBF-FD method in solving Poisson’s equa-
tion with the idea of varying the order of the augmenting
monomials to maintain the global order of convergence over
the domain regardless of the potential variations in the spa-
tial discretisation distance. It should also be noted that some
authors reported p-adaptive methods by locally increasing
the number of shape functions, changing the interpolation
basis functions, or simply increasing the stencil size [46–48].
These approaches are all to some extent p-adaptive, but not
in their true essence. The authors of [49] have introduced a
p-refinement with spatially variable local approximation order
and come closest to a true p-adaptive solution procedure on
scattered nodes. However, this work lacks an automated mark-
ing and refinement strategy for the local approximation order,
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

1029Engineering with Computers (2024) 40:1027–1047
1 3
e.g. based on an error indicator. The automated marking and
refinement strategies were used with the weak form h-padap-
tive clouds [50], where the authors use grid-like h-enrichment
to improve the local field description.
In this paper, we present our attempt to implement the
hp-adaptive strong form mesh-free solution procedure using
the mesh-free RBF-FD approximation on scattered nodes.
Our solution procedure follows a well-established paradigm
based on an iterative loop. To estimate the accuracy of the
numerical solution, we employ original IMEX error indica-
tor. The marking strategy used in this work is based on the
Texas Three Step algorithm [34], where the basic idea is
to estimate the smoothness or analyticity of the numerical
solution. Our refinement strategy is based on the recommen-
dations of [10], where the authors were able to obtain satis-
factory results using a purely h-adaptive solution procedure
for elasticity problems. Although the chosen refinement and
marking strategies are not optimal [36], the obtained results
clearly outperform the non-adaptive approach.
2 hp‑adaptive solution procedure
In the present work, we focus on the implementation of
mesh-free hp-adaptive refinement, which combines the
advantages of h- and p-refinement procedures. The proposed
hp-adaptive solution procedure follows the well-established
paradigm based on an iterative loop, where each iteration
step consists of four modules:
1. Solve – A numerical solution
u
is obtained.
2. Estimate – An error indication of the obtained numeri-
cal solution.
3. Mark – Marking of nodes for refinement/de-refinement.
4. Refine – Refinement/de-refinement of the spatial discre-
tisation and local approximation order of the numerical
method.
The workings of each module are further explained in the
following subsections, while a full hp-adaptive solution
procedure algorithm is given in Algorithm1. For clarity,
Fig.1 also graphically sketches the ultimate goal of a single
refinement iteration.
Algorithm 1 hp -adaptive solution procedure
Input:
The problem, computationaldomain Ω, initialnodal densityfunctio
n
h
:Ω→R, initial approximation orderdistribution m:Ω→N, the maxima
l
nu
mber of iterations Imax and adaptivityparameters αh,p ,β
h,p,λ
h,p,ϑ
h,p.
Output:
The hp-refinednumerical solutionof the problem.
1: function adaptive solve(problem,Ω,h,m,I
max,α
h,p,β
h,p,λ
h,p,ϑ
h,p)
2: for i←0to Imax do
3: Ω←discretise(Ω,h)Discretises domain usingnodal densit
y
function h.
4: solution ←solve(problem,Ω,m)Obtains anumerical solutio
n
to the problem.
5: indicator ←imex(problem,solution,Ω,m)Errorindicato
r
computation.
6: if stopping criteria then
7: return solution
8: endif
9: h, m ←adapt(indicator,h,m,Ω,α
h,p,β
h,p,λ
h,p,ϑ
h,p)
Refine
the nodes and approximation orders.
10: end for
11: return solution
12: end function
2.1 The SOLVE module
First, a numerical solution
u
to the governing problem must
be obtained. In general, the numerical treatment of a sys-
tem of PDEs is done in several steps. First, the domain is
Fig. 1 A sketch of a single hp-refinement iteration for a two-dimen-
sional problem. Note that the exponentially strong source (marked
with red cross) is set at p=
(
1
2,1
3
)
. The refined state has been
obtained by employing h- and p-refinement strategies, thus the num-
ber of nodes and the local approximation orders in the neighbourhood
of the strong source have been modified. Closed form solution has
been used to indicate the error in the estimate module
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

1030 Engineering with Computers (2024) 40:1027–1047
1 3
discretised by positioning the nodes, then the linear differen-
tial operators in each computational node are approximated,
and finally the system of PDEs is discretised and assembled
into a sparse linear system. To obtain a numerical solution
u
, the sparse system is solved.
2.1.1 Domain discretisation
While traditional mesh-based methods discretise the domain
by building a mesh, mesh-free methods simplify this step to
the positioning of nodes, as no information about internodal
connectivity is required. With the mathematical formula-
tion of the mesh-free methods being dimension-independent,
we accordingly choose a dimension-independent algorithm
for node generation based on Poisson disc sampling [51].
Conveniently, the algorithm also supports spatially variable
nodal densities required by the h-adaptive refinement meth-
ods. An example of a variable node density discretisation
can be found in Fig.2.
Interested readers are further referred to the original paper
[51] for more details on the node generation algorithm, its
stand-alone C++ implementation in the Medusa library [52],
and follow-up research focusing on its parallel implementa-
tion [53] and parametric surface discretisations [54].
2.1.2 Approximation oflinear differential operators
Having discretised the domain, we proceed to the approxi-
mation of linear differential operators. In this step, a linear
differential operator
L
is approximated over a set of neigh-
bouring nodes, commonly referred to as stencil nodes.
To derive the approximation, we assume a central point
xc∈Ω
and its stencil nodes
{
x
i}n
i=1
=N for stencil size n.
A linear differential operator in
xc
is then approximated
over its stencil with the following expression
for an arbitrary function u and yet to be determined weights
w
which are computed by enforcing the equality of approxi-
mation(1) for a chosen set of basis functions.
In this work, we use Radial Basis Functions (RBFs)
augmented with monomials. To eliminate the dependency
on a shape parameter, we choose Polyharmonic Splines
(PHS) [14] defined as
for Eucledian distance r. The chosen approximation basis
effectively results in what is commonly called the RBF-FD
approximation method [41].
Furthermore, it is necessary that the stencil nodes
form a so-called polynomial unisolvent set [55]. In this
work, we follow the recommendations of Bayona [14] and
define the stencil size as twice the number of augmenting
monomials, i.e.
for monomial order m and domain dimensionality d. This,
in practice, results in large enough stencil sizes to satisfy
the requirement, so that no special treatment was needed to
assure unisolvency. While special stencil selection strate-
gies showed promising results [11, 56], a common choice
for selecting a set of stencil nodes
N
is to simply select the
nearest n nodes. The latter approach was also used in this
work. Figure2 shows example stencils for different approxi-
mation orders m on domain boundary and its interior.
It is important to note that the augmenting monomials
allow us to directly control the order of the local approxi-
mation method. The approximation order corresponds to
the highest augmenting monomial order m in the approx-
imation basis. However, the greater the approximation
order the greater the computational complexity due to
larger stencil sizes [13]. Nevertheless, the ability to con-
trol the local order of the approximation method sets the
foundation for the p-adaptive refinement.
To conclude the solve module, the PDEs of the govern-
ing problem are discretised and assembled into a global
(1)
(
Lu)(xc)≈
n
∑
i=1
wiu(xi)
,
(2)
f(r)=
{
r
k
,kodd
rklog r,keven
,
(3)
n
=2
(
m+d
m
)
Fig. 2 An example of domain discretisation with scattered nodes and
variable node density. Example stencils are also shown for different
approximation orders m on the domain boundary and its interior
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

1031Engineering with Computers (2024) 40:1027–1047
1 3
sparse system. The solution of the assembled system
stands for the numerical solution
u
.
2.2 The ESTIMATE module (Implicit‑Explicit error
indicator)
In the estimation step, critical areas with high error of the
numerical solution are identified. Identifying such areas is
not a trivial task. In rare cases where a closed form solution
to the governing problem exists, we can directly determine
the accuracy of the numerical solution. Therefore, other
objective metrics, commonly referred to as error indicators,
are needed to indicate areas with high error of the numerical
solution.
2.2.1 IMplicit‑EXplicit (IMEX) error indicator
In this work we will use an error indicator based on the
implicit-explicit [57] evaluation of the considered field.
IMEX makes use of the implicitly obtained numerical solu-
tion and explicit operators (approximated by a higher order
basis) to reconstruct the right-hand side of the governing
problem. To explain the basic idea of IMEX, let us define
a PDE of type
where
L
is a differential operator applied to the scalar field
u and
fRHS
is a scalar function. To obtain an error indica-
tor field
𝜂
, the problem(4) is first solved implicitly using a
lower order approximation
Lim
of operators
L
, obtaining the
solution
uim
in the process. The explicit high order operators
Lex
are then used over the implicitly computed field
uim
to
reconstruct the right-hand side of the problem(4) obtaining
fex
RHS
in the process. The error indication is then calculated
as
𝜂=|fRHS −fex
RHS|
. The calculation steps of the IMEX error
indicator are also shown in Algorithm2.
Algorithm 2 IMEX errorindicator
Input:
The problem, domainΩ,differentialoperators L,low-order approxi-
mation
basis ξ,high order approximationbasis ζ.
Output:
Error indicator field η.
1: function indicate error(problem,Ω,L,ξ,ζ)
2: Lim ←approximate(Ω,ξ)Obtain low-order approximation of
differential operators L.
3: uim,f
RHS←solve(problem,Ω,Lim)Obtain anumerical solution
to the problem.
4: Lex ←approximate(Ω,ζ)Obtain high orderapproximation of
differential operators L.
5: fex
RHS←evaluate(problem,Ω,Lex,u
im)Explicit re-evaluation.
6: η←compute(fRHS,fex
RHS)Obtain error indicatorfield.
7: return η
8: end function
The assumption that the deviation of the explicit high
order evaluation
Lexuim
from the exact
fRHS
corresponds
(4)
Lu=fRHS,
to the error of the solution
uim
is similar to the reasoning
behind the ZZ-type indicators, where the deviation of the
recovered high order solution from the computed solution
characterises the error. As long as the error in
uim
is high,
the explicit re-evaluation will not correctly solve the Equa-
tion(4). However, as the error in
uim
decreases, the differ-
ence between
fRHS
and
fex
RHS
will also decrease, assuming
that the error is dominated by the inaccuracy of
uim
and not
by the differential operator approximation.
It is worth noting that the definition of IMEX is general
in the sense that computing the error indication
𝜂
does
not distinguish between the interior and boundary nodes.
In the boundary nodes, the error indicator
𝜂
is calculated
in the same way as in the interior nodes. In the case of
Dirichlet boundary conditions, the error indicator is triv-
ial because the solution fields are exactly imposed, i.e.
the error indicator results in
𝜂=0
. However, in case of
boundary conditions involving the evaluation of deriva-
tives (Robin and Neumann),
𝜂≠0
.
2.3 The MARK module
After the error indicator
𝜂
has been obtained for each
computational point in domain
Ω
, a marking strategy is
applied. The main goal of this module is to mark the nodes
with too high or too low values of the error indicator to
achieve a uniformly distributed accuracy of the numerical
solution and to reduce the computational cost of the solu-
tion procedure – by avoiding fine local field descriptions
and high order approximations where this is not required.
Moreover, the marking strategy not only decides whether
or not (de-)refinement should take place at a particular
computational node, but also defines the type of refinement
procedure if there are several to choose from. In this work,
we use a modified Texas Three Step marking strategy [30,
58], originally restricted to refinement (no de-refinement)
with the h- and p-refinement types. This chosen strategy
was also considered in one of the recent papers by Eib-
ner [34], who showed that, although extremely simple to
understand and implement, it can provide results good
enough to demonstrate the advantages of mesh-based hp-
adaptive solution procedures.
In each iteration of the adaptive procedure, the marking
strategy starts by checking the error indicator values
𝜂i
for
all computational nodes in the domain. Unlike the originally
proposed marking strategy [34] that used only refinement,
we additionally introduce de-refinement. Therefore, if
𝜂i
is
greater than
𝛼𝜂max
for the maximum indicator value
𝜂max
and
a free model parameter
𝛼∈(0, 1)
, the node is marked for
refinement. If
𝜂i
is less than
𝛽𝜂max
for a free model parameter
𝛽∈(0, 1)∧𝛽≤𝛼
, the node is marked for de-refinement.
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

1032 Engineering with Computers (2024) 40:1027–1047
1 3
Otherwise, the node remains unmarked, which means that
no (de-)refinement should take place. The marking strategy
can be summarised with a single equation
In the context of mesh-based methods, it has already been
observed, that such marking strategy, although easy to
implement, is far from optimal [2, 34]. Additionally, it has
also been demonstrated that in case of smooth solutions
p-refinement is preferred while h-refinement is preferred in
volatile fields, e.g. in vicinity of a singularity in the solution
[2, 36], which cannot be achieved with the chosen marking
strategy. Additional discussion on this issue can be found
in Sect.4, where problems with singularity in the solution
are discussed, and in Sect.2.6.3 where we discuss some
guidelines for possible work on improved marking strategies.
Since our work is focused on the implementation of hp-
adaptive solution procedure rather than discussing the opti-
mal marking strategy, we decided to secure full control over
the marking strategy by treating h- and p-methods separately
– but at the cost of higher number of free parameters. There-
fore, the marking strategy is modified by introducing param-
eters
{𝛼h
,
𝛽h}
and
{𝛼p
,
𝛽p}
for separate treatment of h- and
(5)
⎧
⎪
⎨
⎪
⎩
𝜂i> 𝛼𝜂max, refine
𝛽𝜂max ≤𝜂i≤𝛼𝜂max , do nothing
𝜂i< 𝛽𝜂max, de-refine
.
p-refinements, respectively (see Fig.3 for clarification).
Note that the proposed modified marking strategy can mark
a particular node for h-, p- or hp-(de-)refinement if required,
otherwise the computational node is left unchanged.
2.4 The REFINE module
After obtaining the list of nodes marked for modification, the
refinement module is initialised. In this module, the local field
description and local approximation order are left unchanged
for the unmarked nodes, while the remaining nodes are further
processed to determine other refinement-type-specific details
– such as the amount of the (de-)refinement. Our h-refinement
strategy is inspired by the recent h-adaptive mesh-free solution
of elasticity problem [10], where the following h-refinement
rule was introduced
for the dimensionless parameter
𝜆∈[1, ∞)
allowing us to
control the aggressiveness of the refinement – the larger the
value, the greater the change in nodal density, as shown in
Fig.4 on the left. This refinement rule also conveniently
refines the areas with higher error indicator values more than
(6)
h
new
i(p)=
h
old
i
𝜂i−𝛼𝜂max
𝜂max−𝛼𝜂max (
𝜆−1
)
+
1
Fig. 3 A visual representation
of h- and p-(de)refinement
marking strategy
Fig. 4 A visual representation
of the (de-)refinement strategies
for different values of refine-
ment aggressiveness
𝜆
and
de-refinement aggressiveness
𝜗
. Notice that both refinement
types also have lower and upper
limits
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

1033Engineering with Computers (2024) 40:1027–1047
1 3
those closer to the upper refinement threshold
𝛼h𝜂max
. Simi-
larly, a de-refinement rule is proposed
where parameter
𝜗∈[1, ∞)
allows us to control the aggres-
siveness of de-refinement.
The same refinement(6) and de-refinement(7) rules are
applied to control the order of local approximation (p-refine-
ment), except that this time the value is rounded to the near-
est integer, as shown in Fig.4 on the right. Similarly, and for
the same reasons as for the marking strategy (see Sect.2.3),
we consider a separate treatment of h- and p-adaptive pro-
cedures by introducing (de-)refinement aggressiveness
parameters
{
𝜆
h
,𝜗
h}
and
{
𝜆
p
,𝜗
p}
for h- and p-refinement
types respectively.
2.5 Finalization step
Before the 4 modules can be iteratively repeated, the
domain is re-discretised taking into account the newly
computed local internodal distances
hnew
i(p)
and the local
(7)
h
new
i(p)=
h
old
i
𝛽𝜂max−𝜂i
𝛽𝜂
max
−𝜂
min (
1
𝜗−1
)
+1
,
approximation orders
mnew
i(p)
. However, both are only
known in the computational nodes, while global functions

h
new
(p)
and
mnew(p)
over our entire domain space
Ω
are
required.
We use Sheppard’s inverse distance weighting interpo-
lation using the closest
nh
s
neighbours to construct

h
new
(p)
and the closest
nm
s
neighbours to construct
mnew(p)
. In
general, the proposed refinement strategy can introduce
aggressive and undesirable local jumps in node density,
which ultimately leads to a potential violation of the quasi-
uniform internodal spacing requirement within the stencil.
To mitigate this effect, we use relatively large
nh
s
=
30
to
smoothen such potential local jumps. The
mnew(p)
is much
less sensitive in this respect and therefore a minimum
nm
s
=3
is used.
Figure5 schematically demonstrates 3 examples of
hp-refinements. For demonstration purposes, the refine-
ment parameters for h- and p-adaptivity are the same, i.e.
{
𝛼,𝛽,𝜆,𝜗}=
{
𝛼
h
,𝛽
h
,𝜆
h
,𝜗
h}
=
{
𝛼
p
,𝛽
p
,𝜆
p
,𝜗
p}
. Addition-
ally, the de-refinement aggressiveness
𝜗
and the lower
threshold
𝛽
are kept constant, so that effectively only the
upper limit of refinement
𝛼
and the refinement aggres-
siveness
𝜆
are altered. We observe that the effect of the
Fig. 5 Demonstration of
hp-refinement for selected
values of refinement parameters.
The top left figure shows the
numerical solution before its
refinement, while the rest show
its refined state for different
values of refinement parameters.
Contour lines are used to show
the absolute error of the numeri-
cal solution. To denote the
p-refinement, the nodes are
coloured according to the local
approximation order. For clarity,
all figures are zoomed to show
only the neighbourhood of an
exponentially strong source
e
−a
‖
x−xs
‖2
positioned at
x
s
=
(
1
2
,1
3)
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

1034 Engineering with Computers (2024) 40:1027–1047
1 3
refinement parameters is somewhat intuitive. The greater
the aggressiveness
𝜆
, the better the local field description
and the greater the number of nodes with high approxima-
tion order. A similar effect is observed when manipulating
the upper refinement threshold
𝛼
, except that the effect
comes at a smoother manner. Note also that all refined
states were able to increase the accuracy of the numerical
solution from the initial state.
2.6 Note onmarking andrefinement strategies
With the chosen marking and refinement strategies, a sepa-
rate treatment of h- and p-refinement types turned out to be
a necessary complication for a better overall performance
of the solution procedure. Nevertheless, we have tried to
simplify the solution procedure as much as possible. In the
process, important observations have been made – some of
which we believe should be highlighted. This section there-
fore opens a discussion on important remarks related to the
proposed marking and refinement modules.
2.6.1 The error indicators
Since the h- and p-refinements are conceptually different,
our first attempt was to employ two different error indicators
– one for each type of refinement. We employed the previ-
ously proposed variance of field values [10] for marking the
h-refinement and the approximation order based IMEX for
the p-refinement. Unfortunately, no notable advantages of
such solution procedure has been observed and was therefore
discarded due to the increased implementation complexity.
However, other combinations that might show more promis-
ing results should be considered in future work.
2.6.2 Free parameters
In the proposed solution procedure, each adaptivity type
comes with 4 free parameters that need to be defined, i.e.
{
𝛼
h,p
,𝛽
h,p
,𝜆
h,p
,𝜗
h,p}
. This gives a total of 8 free parameters
that can be fine-tuned to a particular problem. While we have
tried to avoid any kind of fine-tuning, we have nevertheless
observed that these parameters can have a crucial impact on
the overall performance of the hp-adaptive solution proce-
dure in terms of (i) the achieved accuracy of the numerical
solution, (ii) the spatial variability of the error of the numeri-
cal solution, (iii) the computational complexity, and (iv) the
stability of the solution procedure.
We observed that if the refinement aggressiveness
𝜆h
is
too high, the number of nodes can either diverge into unrea-
sonably large domain discretisations or ultimately violate
the quasi-uniform internodal spacing requirement, making
the solution procedure unstable. Note that here we refer to
the stability of the solution of the discretised PDEs, which
ultimately governs the stability of the whole solution pro-
cedure. Furthermore, a large number of nodes combined
with high approximation orders can lead to unreasonably
high computational complexity in a matter of few iterations.
However, when refinement aggressiveness
𝜆h
and
𝜆p
is set
too low, the number of required iterations can increase to
such an extent that the entire solution procedure becomes
inefficient. On top of that, the lower and upper threshold
multipliers
𝛼
and
𝛽
also play a crucial role. If
𝛼
is too low,
almost the entire domain is refined. Moreover, if
𝛼
is too
large, almost no refinement takes place and if it does, it is
extremely local, which again has no beneficial consequences
as it often leads to a violation of the quasi-uniform nodal
distribution requirement.
In our tests, based on extensive experimental parameter
testing, we have selected a reasonable combination of all
8 parameters that lead to a stable solution procedure while
demonstrating the advantages of the proposed hp-adaptive
approach. A thorough analysis of these parameters and their
correlation would most likely lead to better results, as there
is no guarantee that the selected parameters are optimal.
However, such an analysis is beyond the scope of this paper,
whose aim is to present an hp-adaptive solution procedure
in the context of mesh-free methods and not to discuss the
optimal marking and refinement strategies. Nevertheless, we
have tried to reduce the number of free parameters using the
same values for h- and p-adaptivity (see Fig.5). While this
approach also yielded satisfactory results that outperformed
the numerical solutions obtained with uniform nodal and
approximation order distributions in terms of accuracy, the
full 8-parameter formulation easily yielded significantly bet-
ter results.
2.6.3 A step beyondtheartificial refinement strategies
As discussed in Sect.2.3 and later in Sect.4, the Texas
Three Step based marking strategy cannot assure the opti-
mal balance of h- and p-refinements due to missing local
data regularity estimation [2]. In FEM, local Sobolev regu-
larity estimate is commonly used to choose between the h-
and the p-refinement [32–34]. Using an estimate for upper
error bound [59, 60] one could generalise this approach to
meshless methods, essentially upgrading the strategy with
an information on the minimal internodal spacing required
for local approximation of the partial differential operator
of a certain order.
The refinement strategy could also be based on a spe-
cific knowledge about convergence rates and computational
complexity in terms of internodal distance
h(p)
and local
approximation orders
m(p)
.
It has already been shown by Bayona [61] that the approx-
imation error of mesh-free interpolant F is bounded by
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

1035Engineering with Computers (2024) 40:1027–1047
1 3
Note that the constant C present in Equation(8) depends on
the stencil and on the approximation order, both of which
are modified by the hp-adaptive solution procedure. Nev-
ertheless, for the purpose of illustrating how a better mark-
ing strategy could be constructed, we decide to simplify the
Equation(8) to saying that the error e is proportional to
h(p)m(p)
. Knowing the target error
et
, we write the ratio of
et∕e0
as
where
mt
is used to denote the target approximation order
and
m0
is the current order of the approximation used to
compute current error
e0
.
From Equation(9) a smarter guess for target local approx-
imation order can be obtained
Such strategy would conveniently leave the approximation
order unchanged when
et=e0
, increase it when
et<e0
and
decrease it when
et>e0
.
A step even further could be to additionally consider the
change in computational complexity, similar to what the
authors of [35] and [45] have already shown. Therefore, we
believe that future work should consider the minimum local
computational complexity criteria. A rough computational
complexity can be obtained with the help of
for domain dimensionality d and target and current inter-
nodal distances
ht
and
h0
respectively.
2.7 Implementation note
The entire hp-adaptive solution procedure from Algo-
rithm1 is implemented in C++. All meshless methods and
approaches used in this work are included in our in-house
developed Medusa library [52]. The code1was compiled
using g++ (GCC) 9.3.0 for Linux with -03 -DNDEBUG
-fopenmp flags.Post-processing was done using Python 3.10
(8)
‖
F(p)−u(p)
‖
∞≤Ch
m+1
max
p∈Ω �
L
(m+1)
(u(p))
�.
(9)
e
t
e0
∝
hm
t
hm0
=hmt−m0
,
(10)
m
t=m0+ln
e
t
e0
.
(11)
𝜒
∝

mt+d
d
3
1
ht

d

m0+d
d

3

1
h0

d
,
and Jupyter notebooks, also available in the provided git
repository.
3 Demonstration onexponential peak
problem
The proposed hp-adaptive solution procedure is first dem-
onstrated on a synthetic example. We chose a 2-dimensional
Poisson problem with exponentially strong source positioned
at xs
=
(
1
2,1
3
)
. This example is categorized as a difficult
problem and is commonly used to test the performance of
adaptive solution procedures [2, 29, 42, 62]. The problem
has a tractable solution
u(x)=e
−a
‖
x−xs
‖2
, which allows us
to evaluate the precision of the numerical solution
u
, e.g. in
terms of the infinity norm
Governing equations are
for a d-dimensional domain
Ω
and strength
a=103
of the
exponential source. The domain boundary is split into two
sets: Neumann
Γ
n=
{
x,x≤1
2
}
and Dirichlet
Γ
d=
{
x,x>1
2
}
boundaries. An example hp-refined numer-
ical solution is shown in Fig.6.
In the continuation of this paper, the numerical solution
of the final linear system is obtained by employing BiCG-
STAB solver with a ILUT preconditioner from the Eigen
C++ library [63]. Global tolerance was set to
10−15
with a
maximum number of 800 iterations and drop-tolerance and
fill-factor set to
10−5
and 50 respectively. While the initial
adaptivity solution was obtained without the guess, all other
iterations used the previous numerical solution
ui−1
as the
guess for new numerical solution
ui
, effectively reducing
the number of iterations required by the BiCGSTAB solver.
3.1 Convergence analysis ofunrefined solution
The problem is first solved on a two-dimensional unit disc
without employing any refinement procedures, i.e. with
uniform nodal and approximation order distributions. The
shapes approximating the linear differential operators are
(12)
e
∞=
‖
u−u
‖
∞
‖
u
‖∞
,
‖
u
‖
∞=max
i=1,…,N
�
ui
�.
(13)
∇
2u(x)=2ae−a
‖
x−xs
‖2
(2a
�
�
x−x
s�
�
−d)in Ω
,
(14)
u(x)=e
−a
‖
x−xs
‖2
on Γd,
(15)
∇u(x)=−2a(x−xs)e
−a
‖
x−xs
‖2
on Γn,
1 The source code is available at: https:// gitlab. com/ e62Lab/ public/
2022_p_ hp- adapt ivity under tag v1.2.
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

1036 Engineering with Computers (2024) 40:1027–1047
1 3
computed using the RBF-FD with PHS order
k=3
and
monomial augmentation
m∈{2, 4, 6, 8}
.
Figure7 shows the results. Each plotted point is an aver-
age obtained after 50 consecutive runs with slightly different
domain discretisations (a random seed for generating expan-
sion candidates was changed, see [51] for more details). In
this way, we can not only study the convergence behav-
iour, but also evaluate how prone the numerical method is
to non-optimal domain discretisations. The convergence
of the numerical solution for selected monomial augmen-
tations is shown on the left. We observe that due to the
strong source, the convergence rates no longer follow the
theoretical prediction of being proportional to
hm
. Instead,
the convergence rates for a small number of computational
nodes (
N⪅2000
) are significantly lower than that obtained
for larger domain discretisations (
N⪆3000
) for all approxi-
mation orders
m>2
. Furthermore, the accuracy gain using
higher order approximations with small domain discreti-
sations is practically negligible. However, when the local
field description is sufficient, both the numerical solution
and the IMEX error indicator (Fig.7 on the right) give reli-
able results. While we could have forced at least one node in
the neighbourhood of the source, we do not use any special
techniques in this work. Instead, further research is simply
limited to sufficiently large domains so that this observation
does not represent an issue.
Moreover, the behaviour of the IMEX error indicator is
studied on the right side of Fig.7. Here, the approxima-
tion order m means that the implicit numerical solution
uim
was obtained with approximation order m, while the explicit
operators
Lex
from IMEX were approximated using mono-
mials up to and including order
m+2
. The observations
show that the maximum value of the error indicator also
converges with the number of computational nodes. Moreo-
ver, we can also observe the aforementioned change in the
convergence rate of the numerical solution, since the maxi-
mum value of the error indicator for domain sizes
N⪅3000
is approximately constant.
3.2 Analysis ofhp‑refined solution
The same problem is now solved by employing the hp-
adaptivity. Free parameters are adjusted to each refinement
type, as can be seen in Table1. Adaptivity iteration loop is
stopped after a maximum of
Niter
iterations. For practical
use, other stopping criteria could also be used, e.g. based on
the maximum error indicator reduction
for the iteration index j. The shapes are computed with RBF-
FD using the PHS with order
k=3
and local monomial aug-
mentation restricted to choose between approximation orders
m∈{2, 4, 6, 8}
. Note that the IMEX error indicator increases
the local approximation order by 2, effectively using mono-
mial orders
mIMEX ∈{4, 6, 8, 10}
. Furthermore, to avoid
unreasonably large number of computational nodes, the
maximum number of allowed nodes
Nmax
is defined. Once
this number is reached, further h-refinement is prevented and
(16)
𝜂j
max
𝜂
0
max
≤𝛾
,
Fig. 6 Example hp-refined solution to exponential peak problem
Fig. 7 Convergence of unrefined numerical solution (left) and IMEX
error indicator (right). Figure only shows a median value after 50
runs with slightly different domain discretisations. Note that, the
approximation order m in the right figure denotes the approximation
order used to obtain the numerical solution, while the explicit opera-
tors employed by the IMEX error indicator are approximated with
orders
m+2
Table 1 Adaptivity parameters
used to obtain solution to the
peak problem
𝛽h
𝛼h
𝜆h
𝜗h
𝛽p
𝛼p
𝜆p
𝜗p
hmax
Nmax
Niter
0.175 0.225 2.625 1.01
10
−
4
0.05 5 1.258 0.1
2.5 ⋅105
70
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

1037Engineering with Computers (2024) 40:1027–1047
1 3
only de-refinement is allowed, while the p-adaptive method
retains its full functionality. To avoid insufficient local field
description, the local nodal density is limited by an upper
bound, i.e.
h(p)≤hmax
. The order of the PHS is left constant.
3.2.1 A brief analysis ofIMEX error indicator
Figure8 shows example indicator fields for the initial itera-
tion, the intermediate iteration, and the iteration that
achieved the best numerical solution accuracy – hereafter
also referred to as the best-performing iteration or simply
the best iteration. The third column shows the IMEX error
indicator. We can see that the IMEX has successfully located
the position of the strong source at
x
s
=
(
1
2
,1
3)
as the high-
est indicator values are seen in its vicinity. Furthermore, the
second column shows that both the accuracy of the numeri-
cal solution and the uniformity of the error distribution were
significantly improved by the hp-adaptive solution proce-
dure, further proving that IMEX can be successfully used as
a reliable error indicator.
The behaviour of IMEX over 70 adaptivity iterations is
also studied in Fig.9. We are pleased to find that the con-
vergence limit of the indicator around iteration
Niter =60
agrees well with the convergence limit of the numerical
Fig. 8 Refinement demonstration. Initial iteration (top row), inter-
mediate iteration (middle row) and best-performing iteration (bottom
row) accompanied with solution error (middle column) and IMEX
error indicator values (right column). The IMEX values for Dirichlet
boundary nodes are not shown. A red cross is used to mark the loca-
tion of the strong peak
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

1038 Engineering with Computers (2024) 40:1027–1047
1 3
solution. This observation also makes the IMEX error
indicator suitable for stopping criteria. Note that, in the
process, the maximum error of the numerical solution has
been reduced by about 9 orders of magnitude, while the
maximum value of the error indicator has been reduced
by about 7 orders of magnitude. In addition, Fig.9 also
shows the number of computational nodes with respect to
the adaptivity iterations.
3.2.2 Approximation order distribution
The iterative adaptive procedure starts by obtaining the
numerical solution of the unrefined problem setup. In this
step, the approximation with the lowest approximation
order, i.e.
m=2
, is assigned to all computational nodes.
Later, the approximation orders are changed according to
the marking and refinement strategies. Figure8 shows the
approximation order distributions for 3 selected adaptivity
iterations. We can observe that the highest approximation
orders are all near the exponentially strong source. Moreo-
ver, due to h-adaptivity, the node density in the neighbour-
hood of the strong source is also significantly increased, i.e.
hmax∕hmin ≈52
in the best-performing iteration.
After applying the p-refinement strategy in the refinement
step, the approximation order in two neighbouring nodes
may differ by more than one. While numerical experiments
with FEM have shown that heterogeneity of polynomial
order in FEM leads to undesired oscillations of the approxi-
mated solution [64], no similar behaviour was observed
in our analyses with our setup using mesh-free methods.
Thus, in contrast to p-FEM, where additional smoothing of
the approximation order takes place within the refinement
module, we have completely avoided such manipulations and
allow the approximation order in two neighbouring nodes to
differ by more than one.
3.2.3 Convergence rates ofhp‑adaptive solution procedure
Finally, the convergence behaviour of the proposed hp-
adaptive solution procedure is studied. In addition to the
convergence of a single hp-adaptive run, Fig.10 shows the
convergences obtained without the use of refinement proce-
dures, i.e. solutions obtained with uniform internodal spac-
ing and approximation orders over the entire domain. The
figure clearly shows that a hp-adaptive solution procedure
was able to significantly improve the numerical solution in
terms of accuracy and computational points required.
As previously discussed by Eibner [34] and Demkowicz
[36], we believe that a more complex marking and refine-
ment strategies would further improve the convergence
behaviour, but already the proposed hp-adaptive solution
procedure significantly outperforms the unrefined solutions.
Specifically, the refined solution is almost 4 orders of mag-
nitude more accurate than the unrefined solution (for the
highest approximation order
m=8
used) at about
104
com-
putational nodes.
4 Application tolinear elasticity problems
In this section we address two problems from linear elastic-
ity that are conceptually different from the exponential peak
problem discussed in Sect.3. While the solution of exponen-
tial peak problem is infinitely smooth, these two problems
both have a singularity in the solution.
In areas of smooth solution, the hp-strategy should favour
p-refinement (assuming that the local discretization is suffi-
cient, as briefly discussed in Sect.3.1), while near the singu-
larity, h-refinement should be preferred [2, 36]. However, the
Texas Three Step based marking strategy used in this paper
Fig. 9 In the top row convergence of IMEX error indicator (blue)
and convergence of numerical solution (red) within 70 iterations
is shown, while the total number of computational nodes is shown
below
Fig. 10 Convergence of the hp-refined solution compared to the con-
vergence of the unrefined solutions
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

1039Engineering with Computers (2024) 40:1027–1047
1 3
cannot trivially achieve this, since the strategy has no knowl-
edge of the smoothness of the solution field. In addition, the
strategy also cannot perform pure h- or pure p-refinement
[34] (see Fig.3), which would be ideal in the limiting situ-
ations. Instead, the strategy used enforces an increase in the
approximation order by its design – even if the solution is not
smooth and even if low-regularity data is being used to con-
struct the approximation. Nevertheless, in our experiments
we observed an increase of the approximation order near the
singularity only in the first few iterations, while the following
iterations were focused on improving the local field descrip-
tion with h-refinement. This observation is also in agreement
with reports from the literature [2, 34], where authors justify
the use of the Texas Three Step marking strategy also for
problems with singularity in the solution.
4.1 Fretting fatigue contact
The application of the proposed hp-adaptive solution pro-
cedure is further expanded to study a linear elasticity prob-
lem. Specifically, we obtain a hp-refined solution to fretting
fatigue contact problem [65] for which no closed form solu-
tion is known.
The problem dynamics is governed by the Cauchy-Navier
equations
with unknown displacement vector
u
, external body force
f
and Lamé parameters
𝜇
and
𝜆
. The domain of interest is a
thin rectangle of width W, length L and thickness D. Axial
traction
𝜎ax
is applied to the right side of the rectangle, while
a compression force is applied to the centre of the rectangle
to simulate contact. The contact is simulated by a compress-
ing force F generated by two oscillating cylindrical pads of
radius R, causing a tangential force Q. The tractions intro-
duced by the two pads are predicted using an extension of
Hertzian contact theory, which splits the contact area into
the stick and slip zones depending on the friction coefficient
𝜇
and the combined elasticity modulus
E
∗−1=
(
1−
𝜈2
1
E1
+
1−
𝜈2
2
E2)
, where
Ei
and
𝜈i
are the Young’s mod-
ulus and the Poisson’s ratios of the sample and the pad,
respectively. The problem is shown schematically in Fig.11
together with the boundary conditions.
Theoretical predictions from [10] are used to obtain the
contact half-width
with normal traction
(17)
(𝜆+𝜇)∇(∇
⋅
u)+𝜇∇2u=f
(18)
a
=2
√
FR
t𝜋E
∗
,
and tangential traction
for
c
=a
√
1−Q
𝜇f defined as the half-width of the slip zone
and
e
=sgn(Q)
a𝜎
ax
4𝜇p
0
is the eccentricity due to axial loading.
Note that the inequalities
Q≤𝜇F
and
𝜎
ax ≤4
(
1−
√
1−
Q
𝜇F
)
must hold for these expressions to be valid.
Plane strain approximation is used to reduce the prob-
lem from three to two dimensions and symmetry along
the horizontal axis is used to further halve the problem
size. Finally,
Ω = [−L∕2, L∕2] × [−W∕2, 0]
is taken as the
domain.
We take
E1=E2=72.1 GPa
,
𝜈1=𝜈2=0.33
,
L=40 mm
,
W=10 mm
,
t=4 mm
,
F=543 N
,
Q=155 N
,
𝜎ax
=100 MPa
,
R=10 mm
and
𝜇=0.3
for the model
parameters. With this setup, the half-contact width a is equal
to
0.2067 mm
, which is about 200 times smaller than the
domain width W. For stability reasons, the 4 corner nodes
were removed after the domain was discretised.
The linear differential operators are approximated with
RBF-FD using the PHS with order
k=3
and local mono-
mial augmentation limited to choose between approximation
orders
m∈{2, 4, 6, 8}
. The PHS order was left constant dur-
ing the adaptive refinement. The hp-refinement parameters
used to obtain the numerical solution are given in Table2.
(19)
p
(x)=

p0

1−x2
a2,

x

<a
0, else
,p0=

FE∗
t𝜋R
,
(20)
q
(x)=







−𝜇p0

1−x2
a2−c
a

1−(x−e)2
c2

,x−e<c
−𝜇p01−x2
a2,c≤

x−e

and

x

≤
a
0, else
Fig. 11 Fretting fatigue contact problem scheme and boundary condi-
tions
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

1040 Engineering with Computers (2024) 40:1027–1047
1 3
Figure12 shows an example of a hp-refined solution
to fretting fatigue problem in the last adaptivity iteration
with
N=46 626
computational nodes. We see that the
solution procedure has successfully located the two criti-
cal points, i.e. the fixed upper left corner with a stress
singularity and the area in the middle of the upper edge
where contact is simulated. Note that the highest stress
values (about 2 times higher) were calculated in the sin-
gularity in the upper left corner, but these nodes are not
shown as our focus is shifted towards the area under the
contact.
4.1.1 Surface traction underthecontact
For a detailed analysis, we consider the surface trac-
tion
𝜎xx
, as it is often used to determine the location of
crack initiation. The surface traction is shown in Fig.13
for 6 selected adaptivity iterations. The mesh-free nodes
are coloured according to the local approximation order
enforced by the hp-adaptive solution procedure. The mes-
sage of this figure is twofold. First, it is clear that the pro-
posed IMEX error indicator can be successfully used in
linear elasticity problems, and second, we find that the hp-
adaptive solution procedure has successfully approximated
the surface traction near the contact. In doing so, the local
field description under the contact has been significantly
improved and the local approximation orders have taken a
non-trivial distribution.
The surface traction in Fig.13 is additionally accompa-
nied with the FEM results on a much denser mesh with more
than 100,000 DOFs obtained with the commercial solver
Abaqus® [65]. To calculate the absolute difference between
the two methods, the mesh-free solution was interpolated to
Abaqus’s computational points using Sheppard’s inverse dis-
tance weighting interpolation with 2 nearest neighbours. We
see that the absolute difference under the contact decreases
with the number of adaptivity iterations and eventually set-
tles at approximately 2% of the maximum difference from
the initial iteration. As expected, the highest absolute dif-
ference is at the edges of the contact, i.e. around
x=a
and
x=−a
, while the difference is even smaller in the rest of
the contact area. The absolute difference between the two
methods is further studied in Fig.14, where the mean of
|𝜎FEM
xx
−
𝜎mesh-free
xx |
under the contact area, i.e.
−a≤x≤a
, is
shown. We observe that the mesh-free hp-refined solution
converges towards the reference FEM solution with respect
to the adaptivity iterations. Moreover, Fig.14 also shows the
number of computational nodes with respect to the adaptiv-
ity iteration.
4.2 The three‑dimensional Boussinesq’s problem
As a final benchmark problem we solve the three-dimen-
sional Boussinesq’s problem, where a concentrated normal
traction acts on an isotropic half-space [66].
The problem has a closed form solution given in cylindri-
cal coordinates r,
𝜃
and z as
where P is the magnitude of the concentrated force,
𝜈
is the
Poisson’s ratio,
𝜇
is the Lamé parameter and R is the Eucle-
dian distance to the origin. The solution has a singularity at
the origin, which makes the problem ideal for treatment with
adaptive procedures. Furthermore, the closed form solution
also allows us to evaluate the accuracy of the numerical
solution.
In our setup, we consider only a small part of
the problem, i.e.
𝜀=0.1
away from the singularity,
as schematically shown in Fig.15. From a numeri-
cal point of view, we solve the Navier–Cauchy Equa-
tion(17) with Dirichlet boundary conditions described
in(21), where the domain
Ω
is defined as a box, i.e.
Ω = [−1, −𝜀] × [−1, −𝜀] × [−1, −𝜀]
.
Although the closed form solution is given in cylindri-
cal coordinate systems, the problem is implemented using
cartesian coordinates. We employ the proposed mesh-
free hp-adaptive solution procedure where the shapes are
computed with RBF-FD using the PHS with order
k=3
and monomial augmentation restricted to choose between
approximation orders
m∈{2, 4, 6, 8}
. Other hp-refinement
related parameters are given in Table3. For the physical
parameters of the problem, the values
P=−1
,
E=1
and
𝜈=0.33
were assumed.
(21)
ur=Pr
4𝜋𝜇
(
z
R3−1−2𝜈
R(z+R)
)
,u𝜃=
0,
uz=P
4𝜋𝜇(2(1−𝜈)
R+z2
R3),
𝜎rr =P
2𝜋(1−2𝜈
R(z+R)−3r2z
R5),
𝜎
𝜃𝜃 =P(1−2𝜈)
2𝜋(z
R3−1
R(z+R)),
𝜎zz =−3Pz3
2𝜋R5,𝜎rz =−
3Prz2
2𝜋R5,
𝜎r𝜃
=0, 𝜎
𝜃z
=0,
Table 2 Adaptivity parameters
used to obtain solution to
fretting fatigue contact problem
𝛽h
𝛼h
𝜆h
𝜗h
𝛽p
𝛼p
𝜆p
𝜗p
hmax
Nmax
Niter
5⋅10
−
5
10
−
4
5 1.05
10
−
3
0.1 4 1.05
2.5 ⋅10
−
4
5⋅105
19
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

1041Engineering with Computers (2024) 40:1027–1047
1 3
It is worth mentioning, that the final sparse system
was solved using BiCGSTAB with ILUT preconditioner
(employed with an initial guess obtained from the previous
adaptivity iteration), where the global tolerance was set to
10−15
with a maximum number of 500 iterations and drop-
tolerance and fill-factor set to
10−6
and 60 respectively.
Other possible choices and their effect on the solution pro-
cedure are further discussed in Sect.4.2.2.
Example hp-refined numerical solution is given in
Fig.16. We can see that the proposed hp-adaptive solution
procedure is sufficiently robust to obtain a good solution
even for three-dimensional problems with singularities.
Additionally, we also observe that the IMEX error indica-
tor successfully identified the singularity, effectively seen
as an increase in the local field description in the neigh-
bourhood of the concentrated force applied at the origin.
Fig. 12 Example hp-refined fretting fatigue contact solution
Fig. 13 Surface traction
under the contact for selected
iteration steps demonstrat-
ing the hp-adaptivity process.
Colours are used to denote the
local approximation orders.
Numerical solution is addi-
tionally compared against the
Abaqus FEM solution, where
the red line is used to denote the
absolute difference between the
two methods. For clarity, the
two dashed green lines show the
edge contact
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

1042 Engineering with Computers (2024) 40:1027–1047
1 3
4.2.1 The von Mises stress alongthebody diagonal
Figure17 shows further evaluation of the hp-refined mesh-
free numerical solution. Here, the von Mises stress at points
near the body diagonal
(−1, −1, −1)
→
(−𝜀,−𝜀,−𝜀)
is cal-
culated for selected 4 adaptivity iterations and compared to
the analytical values in terms of relative error. In addition,
the nodes are coloured according to the local approxima-
tion order enforced by the hp-adaptive solution procedure.
We can see that the highest relative error of approximately
0.3 at the initial state is observed in the neighbourhood of
the origin. In the final iteration, the relative error is reduced
by about an order of magnitude. We also see that the hp-
adaptive solution procedure has found a non-trivial order
distribution and that the number of nodes in the neighbour-
hood of the corner
(−𝜀,−𝜀,−𝜀)
has increased significantly.
A more quantitative analysis of the mesh-free solution
is given in Fig.18 where the
𝓁1
,
𝓁2
and
𝓁∞
error norms
and number of computational nodes vs. adaptivity iteration
are shown. Compared to the initial state, the hp-adaptive
solution procedure was able to achieve a numerical solution
that was almost two orders of magnitude more accurate. In
the process, the number of computational nodes increased
from 10500 in the initial state to about 80000 in the final
iteration. However, it is interesting to observe that with
the configuration from Table3, none of the computational
nodes used the approximation with the highest order allowed
(
m=8
). Instead, in the final iteration, there were 130 nodes
approximated with
m=6
, and 5937 with
m=4
, while the
rest were approximated with the second order. Note that, as
expected, most of the higher order approximations are near
the concentrated force—which is difficult to represent visu-
ally, so we only give the descriptive data.
For reference, we take the h-refined solution by Slak etal.
[10], who were able to reduce the infinity norm error by
about an order of magnitude with
N≈140 000
nodes in the
final iteration. It is perhaps naive to compare this result with
ours, since the authors use different marking and refinement
strategies and, more importantly, a different error indica-
tor. Nevertheless, the infinity norm error of our hp-refined
solution is in the neighbourhood of
10−3
compared to theirs
at approximately
10−2
with almost twice as many com-
putational nodes. We believe our results could be further
improved by fine-tuning the free parameters, but we decided
to avoid such an approach.
4.2.2 Additional discussion onsolving theglobal sparse
system
In all previous sections, we have completely neglected the
importance of solving the global sparse system in the pro-
posed hp-adaptive solution procedure with a suitable solver.
However, inappropriate choice of solver can lead to inac-
curate or even unstable behaviour and, most importantly,
unreasonably large computational cost. To avoid such flaws,
we compared an iterative BiCGSTAB and BiCGSTAB with
ILUT preconditioner with two direct solvers—namely the
SparseLU and the PardisoLU—on a hp-adaptive solution to
the Boussinesq problem, performing 25 adaptivity iterations
with approximately 10,000 initial nodes and 135,000 nodes
after the last iteration. Note that the iterative BiCGSTAB
solver with ILUT preconditioner was employed with an ini-
tial guess obtained from the previous adaptivity iteration.
In addition to the discussed solvers, we also tried the
SparseQR. While its stability and accuracy were compara-
ble to other solvers, its computational cost was significantly
higher and was therefore removed from further analysis and
from the list of potential candidates. For all performed tests
we used the EIGEN linear algebra library [63].
Let us first examine the sparsity patterns of the systems
assembled at different stages of the hp-adaptive process in
Fig.19, where we can see how the system increases in size
and also becomes less sparse due to globally decreasing the
internodal distance hand increasing the approximation order
p. Additionally, the spectra of the matrices are shown in
the bottom row of Fig.19, where we can see that the ratios
between the real and imaginary parts of the eigenvalues are
in good agreement with previous studies [13, 14, 61].
Fig. 14 Mean surface traction difference between the two methods
under the contact area
Fig. 15 Schematic presentation of Boussinesq’s problem
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

1043Engineering with Computers (2024) 40:1027–1047
1 3
Moreover, Fig.20 presents three different views of the
solvers’ performance: (i) the achieved accuracy of the final
solution for different solvers, (ii) the number of iterations
a solver needs to converge, and (iii) the execution times of
each solver, each with respect to the hp-adaptive iterations.
The differences in final accuracy for different solvers are
marginal. Perhaps the BiCGSTAB shows better stability
behaviour (in terms of error scatter) compared to others.
Nevertheless, it is important to observe, that the SparseLU
only works until the 15th iteration with approximately
50000 nodes, at which point our computer ran out of the
available 12 Gb memory, which is to be expected due to the
computational complexity or SparseLU. PardisoLU, on the
other hand, remains stable through all adaptivity iterations.
Generally speaking, the number of iterations BiCG-
STAB needs to converge increases with hp-adaptivity itera-
tions due to the increasing non-zero elements in the global
system. The BiCGSTAB with a ILUT precoditioner shows
similar behaviour, but with approximately 2/3 less iterations
required. Both direct solvers, of course, require only one
“iteration”. Finally, the analysis of the execution time shows
that the PardisoLU solver is by far the most efficient among
all considered candidates.
With all things considered, PardisoLU seems to be the
the best candidate for hp-adaptive solution procedure. How-
ever, the last adaptivity iteration with approximately 115,000
nodes was coincidentally right at the limit of the available
12 Gb RAM memory—using approximately 10.5 Gb. It is
therefore expected that like SparseLU, the PardisoLU would
soon run out of memory for larger domains. To avoid such
problems, we chose to work with a general purpose iterative
BiCGSTAB solver with ILUT preconditioner employed with
an initial guess, since it shows slightly better computational
efficiency than the pure BiCGSTAB and required only 7.5
Gb of RAM for approximately 135,000 nodes in the final
adaptivity iteration.
Table 3 Adaptivity parameters
used to obtain solution to
Boussinesq’s problem
𝛽h
𝛼h
𝜆h
𝜗h
𝛽p
𝛼p
𝜆p
𝜗p
hmax
Nmax
Niter
10
−
3
10
−
3
3.75 1.01
10
−
4
10
−
2
3 1.5 0.04
7⋅104
20
Fig. 16 Example hp-refined
numerical solution to Boussin-
esq’s problem
Fig. 17 Numerical solution compared to analytical solution at the
nodes near the body diagonal
(−1, −1, −1)
→
(−𝜀,−𝜀,−𝜀)
for
selected iterations
Fig. 18 Convergence of numerical solution along with number of
computational nodes
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

1044 Engineering with Computers (2024) 40:1027–1047
1 3
5 Conclusions
In this paper we establish a baseline for hpstrong form
mesh-free analysis. We have formulated and implemented a
hp-adaptive solution procedure and demonstrated its perfor-
mance in three different numerical experiments.
The cornerstone of the presented hp-adaptive method is
an iterative solve–estimate–mark–refine paradigm with the
modified Texas Three Step marking strategy. The h-refine of
the proposed method relies on an advancing front node posi-
tioning algorithm based on Poisson disc sampling, which
enables dimension-independent node generation support-
ing spatially variable density distributions. For the adaptive
order of the method, we exploit an elegant control over the
order of the approximation via the augmenting monomials
in the approximation basis.
We proposed an IMEX error indicator, where the implicit
solution of the problem is processed with the higher order
local explicit representation of PDE at hand, e.g. if the
implicit solution is computed with a second order approxi-
mation, the explicit re-evaluation happens at fourth order.
Our analyses show that the proposed error indicator suc-
cessfully captures main characteristics of error distributions,
which suffices for the proposed iterative adaptivity.
The proposed hp-adaptive solution procedure is first
demonstrated on a two-dimensional Poisson problem with
exponential source and mixed boundary conditions. Further
demonstration focuses on linear elasticity problems. First,
a 2D fretting fatigue problem – a contact problem with
pronounced peaks in the surface stress, and second, a 3D
Boussinesq’s problem with stress singularity. In both cases,
we have demonstrated the advantages of using the proposed
hp-adaptive approach.
Although the hp-adaptivity introduces additional steps in
the solution procedure and is therefore undoubtedly compu-
tationally more expensive per node than the non-adaptive
Fig. 19 Global sparse matrix
plot (top row) and spectra of
the matrices (bottom row) at
three selected iterations of the
hp-adaptive solution proce-
dure. Note that the spectra are
computed for the BiCGSTAB
solver with an ILUT precondi-
tioning using an estimate from
the previous iteration
Fig. 20 Error of the final solu-
tion with respect to the adaptiv-
ity iteration for different solvers
(left), number of solver iteration
per adaptivity iteration (centre)
and solver compute time for
each adaptivity iteration (right)
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

1045Engineering with Computers (2024) 40:1027–1047
1 3
approach, it is essential in problems that exhibit volatilities
in solution in small regions of the domain. For example sin-
gularity in the contact problem require excessively detailed
numerical analysis near the contact compared to the rest
(the bulk) of the domain. Such cases are extremely difficult
(or even impossible) to solve without adaptivity, since the
minimal uniform hand pdistribution required to capture
these volatilities would lead to unreasonably high computa-
tional complexity. In cases of a smooth solution, however,
the benefits of hp-adaptivity in most cases do not justify its
computational overheads.
We are aware that there are many opportunities for
improvement of presented methodology. The IMEX error
indicator needs further clarification. Other error indicators
should also be implemented and tested. During our experi-
ments, we have found that a marking strategy with more free
parameters leads to better accuracy, but is also more diffi-
cult to understand and control and can be case dependent. A
smarter and more effective refinement and marking strate-
gies are certainly part of future work. These should possibly
take into account more information about the method itself,
e.g. the dependence of the computational complexity on the
approximation order, and most importantly local data regu-
larity to choose between pand hrefinement.
One of our goals in future work is also generalisation
of the presented hp-adaptive solution procedure to time-
dependent problems. The most straightforward approach
to achieve that is to granularly adapt hand pthroughout
the simulation. In its simplest form, the proposed hp-adap-
tivity would be performed at each time step, starting with
the hpdistributions of the previous time step and using
the same adaptivity parameters for all time steps. A more
sophisticated approach would also take into account the
desired accuracy during the simulation, resulting in time-
dependent adaptivity parameters. For example, if one is only
interested in a steady state solution, the desired accuracy
would increase with time, reaching its maximum at steady
state. Additionally, to perform proper adaptive analysis, the
time step should also be adaptive, which requires an addi-
tional step in the hp-adaptive solution procedure.
Acknowledgements . The authors acknowledge the financial support
from the Slovenian Research and Innovation Agency (ARIS) research
core funding No.P2-0095, and research projects No. J2-3048 and No.
N2-0275 (joint research project between National Science Centre,
Poland and ARIS, where Polish research group is Funded by National
Science Centre, Poland under the OPUS call in the Weave programme
2021/43/I/ST3/00228. This research was funded in whole or in part
by National Science Centre (2021/43/I/ST3/00228). For the purpose
of Open Access, the author has applied a CC-BY public copyright
licence to any Author Accepted Manuscript (AAM) version arising
from this submission.).
Declarations
Conflict of interest The authors declare that they have no conflict of
interest. All the co-authors have confirmed to know the submission of
the manuscript by the corresponding author.
Open Access This article is licensed under a Creative Commons Attri-
bution 4.0 International License, which permits use, sharing, adapta-
tion, distribution and reproduction in any medium or format, as long
as you give appropriate credit to the original author(s) and the source,
provide a link to the Creative Commons licence, and indicate if changes
were made. The images or other third party material in this article are
included in the article's Creative Commons licence, unless indicated
otherwise in a credit line to the material. If material is not included in
the article's Creative Commons licence and your intended use is not
permitted by statutory regulation or exceeds the permitted use, you will
need to obtain permission directly from the copyright holder. To view a
copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.
References
1. Upadhyay BD, Sonigra SS, Daxini SD (2021) Numerical analysis
perspective in structural shape optimization: a review post 2000.
Adv Eng Softw 155:102,992
2. Mitchell WF, McClain MA (2014) A comparison of hp-adaptive
strategies for elliptic partial differential equations. ACM Trans
Math Softw (TOMS) 41(1):1–39
3. Segeth K (2010) A review of some a posteriori error estimates for
adaptive finite element methods. Math Comput Simul 80(8):1589–
1600. https:// doi. org/ 10. 1016/j. matcom. 2008. 12. 019. https://
www. scien cedir ect. com/ scien ce/ artic le/ pii/ S0378 47540 80042
30. ESCO 2008 Conference
4. Liu GR, Gu YT (2005) An introduction to meshfree methods and
their programming. Springer, Berlin
5. Liu GR (2002) Mesh free methods: moving beyond the finite ele-
ment method. CRC Press, Boca Raton. https:// doi. org/ 10. 1201/
97814 20040 586
6. Zienkiewicz OC, Taylor RL, Zhu JZ (2005) The finite element
method: its basis and fundamentals. Elsevier, Amsterdam
7. van der Sande K, Fornberg B (2021) Fast variable density 3-d
node generation. SIAM J Sci Comput 43(1):A242–A257
8. Shankar V, Kirby RM, Fogelson AL (2018) Robust node genera-
tion for mesh-free discretizations on irregular domains and sur-
faces. SIAM J Sci Comput 40(4):A2584–A2608
9. Jacquemin T, Suchde P, Bordas SP (2023) Smart cloud collo-
cation: geometry-aware adaptivity directly from CAD. Comput
Aided Design 154:103409. https:// doi. org/ 10. 1016/j. cad. 2022.
103409. https:// linki nghub. elsev ier. com/ retri eve/ pii/ S0010 44852
20014 27
10. Slak J, Kosec G (2019) Adaptive radial basis function-generated
finite differences method for contact problems. Int J Numer Meth-
ods Eng 119(7):661–686
11. Davydov O, Oanh DT (2011) Adaptive meshless centres and rbf
stencils for poisson equation. J Comput Phys 230(2):287–304
12. Jacquemin T, Bordas SPA (2021) A unified algorithm for the
selection of collocation stencils for convex, concave, and singular
problems. Int J Numer Methods Eng 122(16):4292–4312
13. Jančič M, Slak J, Kosec G (2021) Monomial augmentation guide-
lines for rbf-fd from accuracy versus computational time perspec-
tive. J Sci Comput 87(1):1–18
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

1046 Engineering with Computers (2024) 40:1027–1047
1 3
14. Bayona V, Flyer N, Fornberg B, Barnett GA (2017) On the role
of polynomials in rbf-fd approximations: Ii. numerical solution of
elliptic pdes. J Comput Phys 332:257–273
15. Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P (1996)
Meshless methods: an overview and recent developments. Comput
Methods Appl Mech Eng 139(1–4):3–47
16. Kosec G, Šarler B (2014) Simulation of macrosegregation with
mesosegregates in binary metallic casts by a meshless method.
Eng Anal Bound Elements 45:36–44. https:// doi. org/ 10. 1016/j.
engan abound. 2014. 01. 016. https:// linki nghub. elsev ier. com/ retri
eve/ pii/ S0955 79971 40002 90
17. Maksić M, Djurica V, Souvent A, Slak J, Depolli M, Kosec G
(2019) Cooling of overhead power lines due to the natural convec-
tion. Int J Electrical Power Energy Syst 113:333–343. https:// doi.
org/ 10. 1016/j. ijepes. 2019. 05. 005. https:// linki nghub. elsev ier. com/
retri eve/ pii/ S0142 06151 83400 55
18. Gui Wz, Babuska I (1985) The h, p and hp versions of the finite
element method in 1 dimension. part 3. the adaptive hp version.
Tech. rep., Maryland Univ College Park Lab for Numerical
Analysis
19. Gui WZ, Babuška I (1986) The h, p and hp versions of the finite
element method in 1 dimension. part ii. the error analysis of the h
and hp versions. Numerische Mathematik 49(6):613–657
20. Devloo12 PR, Bravo CM, Rylo EC (2012) Recent developments
in hp adaptive refinement
21. Zienkiewicz OC, Zhu JZ (1987) A simple error estimator and
adaptive procedure for practical engineerng analysis. Int J Numer
Methods Eng 24(2):337–357
22. González-Estrada OA, Natarajan S, Ródenas JJ, Bordas SP
(2021) Error estimation for the polygonal finite element method
for smooth and singular linear elasticity. Comput Math Appl
92:109–119
23. Thimnejad M, Fallah N, Khoei AR (2015) Adaptive refinement
in the meshless finite volume method for elasticity problems.
Comput Math Appl 69(12):1420–1443. https:// doi. org/ 10. 1016/j.
camwa. 2015. 03. 023
24. Angulo A, Pozo LP, Perazzo F (2009) A posteriori error estima-
tor and an adaptive technique in meshless finite points method.
Eng Anal Bound Elements 33(11):1322–1338. https:// doi. org/
10. 1016/j. engan abound. 2009. 06. 004
25. Oanh DT, Davydov O, Phu HX (2017) Adaptive rbf-fd method
for elliptic problems with point singularities in 2d. Appl Math
Comput 313:474–497
26. Sang-Hoon P, Kie-Chan K, Sung-Kie Y (2003) A posterior error
estimates and an adaptive scheme of least-squares meshfree
method. Int J Numer Methods Eng 58(8):1213–1250. https://
doi. org/ 10. 1002/ nme. 817
27. Afshar M, Naisipour M, Amani J (2011) Node moving adaptive
refinement strategy for planar elasticity problems using discrete
least squares meshless method. Finite Element Anal Design
47(12):1315–1325
28. Guo B, Babuška I (1986) The hp version of the finite element
method. Comput Mech 1(1):21–41
29. Mitchell WF (2016) Performance of hp-adaptive strategies for
3d elliptic problems
30. TinsleyOden J, Wu W, Ainsworth M (1995) In: Modeling, mesh
generation, and adaptive numerical methods for partial differ-
ential equations (Springer), pp 347–366
31. Ainsworth M, Senior B (1997) Aspects of an adaptive hp-finite
element method: adaptive strategy, conforming approxima-
tion and efficient solvers. Comput Methods Appl Mech Eng
150(1–4):65–87
32. Houston P, Senior B, Süli E (2003) In: Numerical mathematics
and advanced applications (Springer), pp 631–656
33. Houston P, Süli E (2005) A note on the design of hp-adaptive
finite element methods for elliptic partial differential equations.
Comput Methods Appl Mech Eng 194(2–5):229–243
34. Eibner T, Melenk JM (2007) An adaptive strategy for hp-fem
based on testing for analyticity. Comput Mech 39(5):575–595
35. Bürg M, Dörfler W (2011) Convergence of an adaptive hp finite
element strategy in higher space-dimensions. Appl Numer Math
61(11):1132–1146
36. Demkowicz L, Rachowicz W, Devloo P (2002) A fully auto-
matic hp-adaptivity. J Sci Comput 17(1):117–142
37. Rachowicz W, Pardo D, Demkowicz L (2006) Fully automatic
hp-adaptivity in three dimensions. Comput Methods Appl Mech
Eng 195(37–40):4816–4842
38. Benito J, Urena F, Gavete L, Alvarez R (2003) An h-adaptive
method in the generalized finite differences. Comput Methods
Appl Mech Eng 192(5–6):735–759
39. Liu G, Kee BB, Chun L (2006) A stabilized least-squares radial
point collocation method (ls-rpcm) for adaptive analysis. Com-
put Methods Appl Mech Eng 195(37–40):4843–4861
40. Hu W, Trask N, Hu X, Pan W (2019) A spatially adaptive high-
order meshless method for fluid-structure interactions. Comput
Methods Appl Mech Eng 355:67–93
41. Tolstykh A, Shirobokov D (2003) On using radial basis func-
tions in a “finite difference mode’’ with applications to elasticity
problems. Comput Mech 33(1):68–79
42. Oanh DT, Tuong NM (2022) An approach to adaptive refine-
ment for the rbf-fd method for 2d elliptic equations. Appl
Numer Math 178:123–154
43. Tóth B, Düster A (2022) h-adaptive radial basis function finite
difference method for linear elasticity problems. Comput
Mech:1–20
44. Fan L (2019) Adaptive meshless point collocation methods:
investigation and application to geometrically non-linear solid
mechanics. Ph.D. thesis, Durham University
45. Mishra PK, Ling L, Liu X, Sen MK (2020) Adaptive radial basis
function generated finite-difference (rbf-fd) on non-uniform
nodes using
p
-refinement. arXiv preprint arXiv: 2004. 06319
46. Milewski S (2021) Higher order schemes introduced to the
meshless fdm in elliptic problems. Eng Anal Bound Elements
131:100–117
47. Albuquerque-Ferreira A, Ureña M, Ramos H (2021) The general-
ized finite difference method with third-and fourth-order approxi-
mations and treatment of ill-conditioned stars. Eng Anal Bound
Elements 127:29–39
48. Liszka T, Duarte C, Tworzydlo W (1996) hp-meshless cloud
method. Computer Methods Appl Mech Eng 139(1–4):263–288
49. Jančič M, Slak J, Kosec G (2021) In: 2021 6th International Con-
ference on Smart and Sustainable Technologies (SpliTech), pp
01–06. https:// doi. org/ 10. 23919/ SpliT ech52 315. 2021. 95664 01
50. Duarte CA, Oden JT (1996) An hp adaptive method using clouds.
Computer Methods Appl Mech Eng 139(1–4):237–262
51. Slak J, Kosec G (2019) On generation of node distribu-
tions for meshless pde discretizations. SIAM J Sci Comput
41(5):A3202–A3229
52. Slak J, Kosec G (2021) Medusa: a c++ library for solving pdes
using strong form mesh-free methods. ACM Trans Math Softw
(TOMS) 47(3):1–25
53. Depolli M, Slak J, Kosec G (2022) Parallel domain discretization
algorithm for RBF-FD and other meshless numerical methods for
solving PDEs. Comput Struct 264:106773. (Publisher: Elsevier)
54. Duh U, Kosec G, Slak J (2021) Fast variable density node genera-
tion on parametric surfaces with application to mesh-free meth-
ods. SIAM J Sci Comput 43(2):A980–A1000
55. Wendland H (2004) Scattered data approximation. Cambridge
Monographs on Applied and Computational Mathematics.
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

1047Engineering with Computers (2024) 40:1027–1047
1 3
Cambridge University Press, Cambridge. https:// doi. org/ 10. 1017/
CBO97 80511 617539
56. Davydov O, Oanh DT, Tuong NM (2023) Improved stencil selec-
tion for meshless finite difference methods in 3d. J Comput Appl
Math 425:115031. https:// doi. org/ 10. 1016/j. cam. 2022. 115031.
https:// www. scien cedir ect. com/ scien ce/ artic le/ pii/ S0377 04272
20062 9X
57. Jančič M, Strniša F, Kosec G (2022) In: 2022 7th International
Conference on Smart and Sustainable Technologies (SpliTech),
pp 01–04. https:// doi. org/ 10. 23919/ SpliT ech55 088. 2022. 98543 42
58. Heuer N, Mellado ME, Stephan EP (2001) hp-adaptive two-level
methods for boundary integral equations on curves. Computing
67(4):305–334
59. Bayona V (2019) An insight into rbf-fd approximations aug-
mented with polynomials. Comput Math Appl 77(9):2337–2353
60. Tominec I, Larsson E, Heryudono A (2021) A least squares radial
basis function finite difference method with improved stability
properties. SIAM J Sci Comput 43(2):A1441–A1471. https:// doi.
org/ 10. 1137/ 20M13 20079
61. Bayona V (2019) Comparison of moving least squares and rbf+
poly for interpolation and derivative approximation. J Sci Comput
81(1):486–512
62. Daniel P, Ern A, Smears I, Vohralík M (2018) An adaptive hp-
refinement strategy with computable guaranteed bound on the
error reduction factor. Comput Math Appl 76(5):967–983
63. Guennebaud G, Jacob B, etal (2010) Eigen v3. http:// eigen. tuxfa
mily. org
64. Wakeni MF, Aggarwal A, Kaczmarczyk L, McBride AT, Atha-
nasiadis I, Pearce CJ, Steinmann P (2022) A p-adaptive, implicit-
explicit mixed finite element method for diffusion-reaction prob-
lems. Int J Numer Methods Eng
65. Kosec G, Slak J, Depolli M, Trobec R, Pereira K, Tomar S, Jac-
quemin T, Bordas SP, Wahab MA (2019) Weak and strong from
meshless methods for linear elastic problem under fretting contact
conditions. Tribol Int 138, 392–402. (Publisher: Elsevier)
66. Slaughter WS (2002) Three-dimensional problems.
Birkhäuser, Boston, pp 331–386. https:// doi. org/ 10. 1007/
978-1- 4612- 0093-2_9
Publisher's Note Springer Nature remains neutral with regard to
jurisdictional claims in published maps and institutional affiliations.
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

1.
2.
3.
4.
5.
6.
Terms and Conditions
Springer Nature journal content, brought to you courtesy of Springer Nature Customer Service Center GmbH (“Springer Nature”).
Springer Nature supports a reasonable amount of sharing of research papers by authors, subscribers and authorised users (“Users”), for small-
scale personal, non-commercial use provided that all copyright, trade and service marks and other proprietary notices are maintained. By
accessing, sharing, receiving or otherwise using the Springer Nature journal content you agree to these terms of use (“Terms”). For these
purposes, Springer Nature considers academic use (by researchers and students) to be non-commercial.
These Terms are supplementary and will apply in addition to any applicable website terms and conditions, a relevant site licence or a personal
subscription. These Terms will prevail over any conflict or ambiguity with regards to the relevant terms, a site licence or a personal subscription
(to the extent of the conflict or ambiguity only). For Creative Commons-licensed articles, the terms of the Creative Commons license used will
apply.
We collect and use personal data to provide access to the Springer Nature journal content. We may also use these personal data internally within
ResearchGate and Springer Nature and as agreed share it, in an anonymised way, for purposes of tracking, analysis and reporting. We will not
otherwise disclose your personal data outside the ResearchGate or the Springer Nature group of companies unless we have your permission as
detailed in the Privacy Policy.
While Users may use the Springer Nature journal content for small scale, personal non-commercial use, it is important to note that Users may
not:
use such content for the purpose of providing other users with access on a regular or large scale basis or as a means to circumvent access
control;
use such content where to do so would be considered a criminal or statutory offence in any jurisdiction, or gives rise to civil liability, or is
otherwise unlawful;
falsely or misleadingly imply or suggest endorsement, approval , sponsorship, or association unless explicitly agreed to by Springer Nature in
writing;
use bots or other automated methods to access the content or redirect messages
override any security feature or exclusionary protocol; or
share the content in order to create substitute for Springer Nature products or services or a systematic database of Springer Nature journal
content.
In line with the restriction against commercial use, Springer Nature does not permit the creation of a product or service that creates revenue,
royalties, rent or income from our content or its inclusion as part of a paid for service or for other commercial gain. Springer Nature journal
content cannot be used for inter-library loans and librarians may not upload Springer Nature journal content on a large scale into their, or any
other, institutional repository.
These terms of use are reviewed regularly and may be amended at any time. Springer Nature is not obligated to publish any information or
content on this website and may remove it or features or functionality at our sole discretion, at any time with or without notice. Springer Nature
may revoke this licence to you at any time and remove access to any copies of the Springer Nature journal content which have been saved.
To the fullest extent permitted by law, Springer Nature makes no warranties, representations or guarantees to Users, either express or implied
with respect to the Springer nature journal content and all parties disclaim and waive any implied warranties or warranties imposed by law,
including merchantability or fitness for any particular purpose.
Please note that these rights do not automatically extend to content, data or other material published by Springer Nature that may be licensed
from third parties.
If you would like to use or distribute our Springer Nature journal content to a wider audience or on a regular basis or in any other manner not
expressly permitted by these Terms, please contact Springer Nature at
onlineservice@springernature.com