Adaptive shape optimization with NURBS designs and PHT-splines for solution approximation in time-harmonic acoustics Javier Videlaa Ahmed Mostafa Shaabanb Elena Atroshchenkoa1

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Adaptive shape optimization with NURBS designs and PHT-splines

for solution approximation in time-harmonic acoustics

Javier Videlaa, Ahmed Mostafa Shaabanb, Elena Atroshchenkoa, 1

aSchool of Civil and Environmental Engineering, University of New South Wales, Sydney, Australia

bInstitute of Structural Mechanics, Bauhaus-Universit¨at Weimar, Weimar, Germany

Abstract

Geometry Independent Field approximaTion (GIFT) was proposed as a generalization of

Isogeometric analysis (IGA), where diﬀerent types of splines are used for the parameterization

of the computational domain and approximation of the unknown solution. GIFT with Non-

Uniform Rational B-Splines (NUBRS) for the geometry and PHT-splines for the solution

approximation were successfully applied to problems of time-harmonic acoustics, where it

was shown that in some cases, adaptive PHT-spline mesh yields highly accurate solutions at

lower computational cost than methods with uniform reﬁnement. Therefore, it is of interest

to investigate performance of GIFT for shape optimization problems, where NURBS are

used to model the boundary with their control points being the design variables and PHT-

splines are used to approximate the solution adaptively to the boundary changes during the

optimization process.

In this work we demonstrate the application of GIFT for 2D acoustic shape optimization

problems and, using three benchmark examples, we show that the method yields accurate

solutions with signiﬁcant computational savings in terms of the number of degrees of freedom

and computational time.

Key words: Geometry Independent Field approximaTion, Helmholtz equation, shape

optimization, NURBS, PHT-splines

1. Introduction

Due to rapid civil and transport development, noise pollution has become an important

public health concern, requiring eﬃcient noise control solutions, such as design of structures

(e.g. noise barriers) with optimal acoustic performance. Numerical design is based on the

solution of the wave propagation problem, modeled by the Helmholtz equation.

Numerical methods for the Helmholtz equation encounter two major challenges: the so-

called “pollution error” and treatment of unbounded domains. The pollution error results

from the numerical dispersion error, which is related to the discrepancy between the exact and

numerical wave number (k). It has been proven theoretically that for 2D and 3D problems,

the pollution error cannot be fully eliminated [1]. It is controlled by adapting the mesh

size to the wave length, i.e. the element size is held a few times smaller than the wave

1Corresponding author, eatroshch@gmail.com, e.atroshchenko@unsw.edu.au

arXiv:2210.04480v1 [cs.CE] 10 Oct 2022

length. Hence, the number of elements grows proportionally to the wave number, making

computations for large kunfeasible. However, pollution error is inversely proportional to

the order of shape functions and hence can be signiﬁcantly reduced by the use of higher-

order approximations. This makes isogeometric analysis (IGA), with its natural feature of

polynomial degree elevation or p−reﬁnement, an attractive alternative for solution of the

Helmholtz equation, as demonstrated in [2,3].

Another source of error in modeling wave propagation in unbounded domains results

from introducing a boundary truncation domain (usually a circle of radius Rin 2D and a

sphere of radius Rin 3D), in which the Sommerfeld radiation condition is modeled by the

Absorbing Boundary Condition (ABC). The accuracy of modeling the ABC increases with R,

nonetheless, larger computational domain leads to a larger system and higher computational

cost. Another approach to reduce domain truncation error is by using higher order ABCs.

This requires higher order derivatives of the shape functions, which present a problem in

linear FEM, but can be easily overcome by the use of higher order NURBS in IGA. A

detailed study of various ABCs in the context of Isogeometric Collocation was performed in

[4].

Isogeometric analysis (IGA) and its subsequent variations have been a topic of active

research since IGA was originally introduced in 2005 by Hughes et al. [5]. The main objec-

tive of IGA is to connect the Computer-Aided Design (CAD) model directly with numerical

analysis. In most cases, Non-Uniform Rational B-Spline (NURBS) are able to preserve the

original geometry replacing Lagrange polynomials in the FEM discretization. It has been

successfully proven that, IGA could be implemented in several engineering applications, such

as structural mechanics [6], ﬂuid-structure interaction [7], fracture mechanics [8], electromag-

netics [9], Helmholtz equation [2,3], among others. Ref. [10] presents a complete review for

the application of IGA in diﬀerent engineering aspects.

The necessity to truncate the boundary presents a serious disadvantage of domain-type

methods in comparison with the boundary-type methods, such as boundary element method

(BEM) and its isogeometric variation, namely IGABEM. In BEM, the Sommerfeld radia-

tion condition at inﬁnity is automatically satisﬁed by the fundamental solutions and the

mesh burden is reduced to the boundary discretization [11,12]. BEM and IGABEM have

been widely applied to acoustic problems in exterior domains [13]. Moreover, IGABEM has

been eﬃciently paired with optimization methods for acoustic shape optimization in diﬀerent

research works [14–17].

One of the fundamental problems of the numerical formulas which utilize either NURBS

or B-splines as basis functions (such as IGA) is that, in 2D and 3D applications, basis

functions are generated as a tensor-product of 1D structures without the ability of local

reﬁnement. This makes them expensive in terms of computational resources. Diﬀerent basis

functions have been proposed to overcome this disadvantage, such as T-splines [18], truncated

hierarchical B-splines (THB-splines) [19] and Polynomial splines over Hierarchical T-meshes

(PHT-Splines) [20,21]. All of the aforementioned basis functions have been eﬃciently applied

to the problems of statics, dynamics, acoustics. See [19,22,23].

Geometry Independent Field approximaTion (GIFT) is a numerical scheme proposed

in [24] as an alternative to iso-parametric methods that decouples the geometry and ﬁeld

spaces. It consists in employing certain basis functions (for example, NURBS) to model the

computational domain, whereas employing diﬀerent basis functions to approximate the ﬁeld.

GIFT with NURBS to model the geometry and PHT-splines to model the unknown solution

were applied to problems of linear elasticity and Laplace equation [24], bending of cracked

Kirchhoﬀ-Love plates [25], and time-harmonic acoustics [23]. It was demonstrated that, the

adaptive local reﬁnement of the solution can obtain optimal convergence rates in the cases

of solutions of reduced continuity. Additionally, Jansari et al. [26] developed the research

work of [23] by enriching the PHT-splines solution with the partition of unity property using

plane waves, showing in some cases that, the enrichment provides an enhancement of several

orders of magnitude in the overall solution error.

In the ﬁeld of optimization, IGA formulations have been applied to many engineering ap-

plications, such as: structural shape optimization [27,28], composite structural optimization

[29], acoustic shape optimization [30], piezoelectric energy harvesters [31,32], thermal meta-

materials [33], heat conduction problems [34] and ﬂuids [35] (for a comprehensive review, the

reader is referred to [36]). In addition to the better accuracy of IGA over FEM per degree

of freedom because of the NURBS higher order and higher continuity, IGA addresses an im-

portant issue in FEM shape optimization associated with the boundary representation and

the evolution of mesh following the changing boundary. In IGA, the set of control variables

that parameterize the boundary is naturally chosen as design variables, providing a tight

link between the design, analysis and optimization models. The same approach to shape

optimization is applicable in GIFT.

The optimization methods are classiﬁed in two families: gradient-free and gradient-based

methods. The family of gradient-free methods includes, for example, Particle Swarm Op-

timization (PSO) [37,38] implemented in shape optimization problems in [15,27], Genetic

Algorithm [39] and its optimization applications in [40], among others. The main advan-

tage of gradient-free methods is their ability to ﬁnd global minimum without any sensitiv-

ity analysis, however, at a much higher computational cost than gradient-based methods.

Gradient-based methods [41,42] have much higher convergence rate, however, depending on

the initial guess, they may converge to a local minimum. The performance of gradient-based

optimization methods can be further improved by providing exact gradients, obtained from

the shape derivatives of the objective function, the weak form of the problem and constraints

to perform the sensitivity analysis [14,28]. However, in many applications shape derivatives

are diﬃcult to obtain. In such case, gradients are calculated using ﬁnite-diﬀerence approx-

imations. This can be done eﬃciently for small and medium-size problems. In this work,

we use Sequential Quadratic Programming (SQP) algorithm, which is considered as one of

the most eﬀective methods for solving nonlinear constraint optimization problems [43]. In

SPQ, the sequence of quadratic sub-problems is solved at each iteration to obtain the search

direction. The algorithm is eﬃciently implemented within Matlab $fmincon$function.

Combining shape optimization with adaptive reﬁnement can be traced back to the 80’s.

For example, the works of Kikuchi et al. [44] and Canales et al. [45], are devoted to study

the shape optimization problems for linear elasticity using FEM and adaptive reﬁnements.

Both works show that it is possible to achieve a process of automatic mesh generation and

shape optimization, while the main drawback is that the mesh is prone to distortion, which

is a usual problem with FE formulations. In a more recent work, Mohite and Upadhyay [46]

proposed an adaptive shape optimization framework for laminated composite plates, showing

the advantage of adaptive reﬁnement over uniform reﬁnement.

As pointed out by the review study from Upadhyay et al. [47], adaptive mesh reﬁnement

is not fully integrated in practical applications, since the mesh reﬁnement requires accesses to

the exact geometry, which means, an automatic communication with CAD model is needed.

The later is a drawback that can be alleviated with our GIFT formulation.

In the literature related to adaptive optimization with splines, Chen et al. [48] proposed

an adaptive shape optimization scheme using T-splines and ﬁnite cell method for struc-

tural shape optimization problems. Topology optimization using morphable components and

Hierarchical B-splines (HB-splines) [49], as well as Truncated Hierarchical B-splines (THB-

Splines) has been introduced by Xie et al.[50] showing superior numerical performance over

its uniform counterpart.

It is of particular interest the work of Gupta et al. [51] on adaptive topology optimization

using GIFT. In their work, they used NURBS to model the geometry, and PHT-splines to

approximate the unknown displacement ﬁeld and the density function in the framework of

Solid Isotropic Material with Penalization (SIMP) method. It was shown that adaptive GIFT

can achieve up to 90% CPU time reduction in comparison with uniform meshes.

In this paper, an adaptive shape optimization scheme for Helmholtz problems is proposed.

GIFT scheme is employed, using PHT-splines to discretize the sound ﬁeld, while NURBS are

utilized to model the boundary and the interior of the computational domain. The shape

optimization is performed over the control points that deﬁne the NURBS boundary, while

adaptive reﬁnement is done over the PHT-spline basis functions. This allows us to reduce

both the number of the degrees of freedom, as well as the computational time of the full

optimization process. The adaptive optimization in this work is based on the recovery-based

error estimator, previously proposed in [23] for the Helmholtz equation and PHT-splines.

The remainder of the paper is organized as follows. Section 2is assigned for the prelimi-

naries, where the boundary value problem for the Helmholtz equation, NURBS, PHT-splines,

GIFT formulation, and a generic shape optimization problem are introduced. Section 2.6 in

particular is devoted to the adaptive optimization. In Section 3, numerical results for three

benchmark examples are discussed. Conclusions are drawn in Section 4.

2. Preliminaries

2.1. Boundary Value Problem (BVP) for the Helmholtz equation

The boundary value problem for the Helmholtz equation, as explained in Figure 1, in

domain Ω ∈R2with boundary Γ consists in ﬁnding the spacial component of the acoustic

pressure, u, such that:

∆u+k2u= 0 in Ω (1a)

u=gon ΓD(1b)

∂u

∂n=ikh on ΓN(1c)

∂u

∂n+αu =fon ΓR(1d)

where ∆ is the scalar Laplace operator, nis a unit normal vector on Γ, outward to Ω, k

is the wave number, and iis the imaginary unit. Dirichlet, Neumann and Robin boundary

conditions are prescribed on parts of the boundary, ΓD,ΓNand ΓRrespectively (Γ = ΓD∪

ΓN∪ΓRand ΓD∩ΓN= ΓD∩ΓR= ΓR∩ΓN=∅), in which g,h,αand fare prescribed

functions. For exterior problems, urefers to a scattered wave and the boundary conditions

on the scatterer boundary are written in terms of the incoming wave uinc and its normal

derivative, [23], i.e.:

u=−uinc on ΓD

∂u

∂n=−∂uinc

∂non ΓN

(2)

Ω

ΓD

ΓN

ΓR

scatterer

Figure 1: Helmholtz acoustic problem.

Additionally, the solution satisﬁes the Sommerfeld radiation condition prescribed at in-

ﬁnity to truncate all possible reﬂections of spurious acoustic waves from the far-ﬁeld. This

condition is formulated for 2D problems as follows [11]:

lim

r→∞ √r ∂u

∂r −iku!= 0.(3)

where ris the distance from the origin. In domain-type methods, the Sommerfeld radia-

tion condition is replaced by the Absorbing Boundary Condition (ABC) on the truncation

boundary Σ (usually given by a circle of radius Rin 2D) [4,52]. In this work we use

Bayliss–Gunzburger–Turkel condition of order 1, referred as BGT1, formulated in polar co-

ordinates (r, θ) as:

∂u

∂r +Bu= 0,(4)

where

Bu=1

2R−iku(5)

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AdaptiveshapeoptimizationwithNURBSdesignsandPHT-splinesforsolutionapproximationintime-harmonicacousticsJavierVidelaa,AhmedMostafaShaabanb,ElenaAtroshchenkoa;1aSchoolofCivilandEnvironmentalEngineering,UniversityofNewSouthWales,Sydney,AustraliabInstituteofStructuralMechanics,Bauhaus-UniversitatWeimar...

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Adaptive shape optimization with NURBS designs and PHT-splines for solution approximation in time-harmonic acoustics Javier Videlaa Ahmed Mostafa Shaabanb Elena Atroshchenkoa1

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