Stochastic-deterministic boundary element modelling of transcranial electric stimulation using a three layer head model

Šušnjara, Anna; Verhnjak, Ožbej; Poljak, Dragan; Cvetković, Mario; Ravnik, Jure

doi:10.1016/j.enganabound.2020.11.010

Cited by 10 publications

(6 citation statements)

References 35 publications

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“…The boundary integral Equation ( 5) is used as the basis of the Boundary Element Method (BEM) solver, which we developed. The numerical implementation is based on our Laplace BEM solver [38,39]. We consider the boundary Γ = ∑ l Γ l to be decomposed into boundary elements Γ l :…”

Section: Boundary Element Solution Of Stokes Flow Over a Particlementioning

confidence: 99%

“…A system of linear equations is set up for all unknowns, where in case of unknown {u x } or {q x } Equation ( 8) is used, in case of unknown {u y } or {q y } Equation ( 9) is used and in case of unknown {u z } or {q z } Equation ( 10) is used. Additional details of the BEM employed, such as implementation of integration, can be found in [38,39].…”

Section: Boundary Element Solution Of Stokes Flow Over a Particlementioning

confidence: 99%

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A Model for Translation and Rotation Resistance Tensors for Superellipsoidal Particles in Stokes Flow

Štrakl

Hriberšek

Wedel

et al. 2022

JMSE

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In this paper, forces and torques on solid, non-spherical, orthotropic particles in Stokes flow are investigated by using a numerical approach on the basis of the Boundary Element Method. Different flow patterns around a particle are considered, taking into account the contributions of uniform, rotational and shear flows, to the force and the torque exerted on the particle. The expressions for the force and the toque are proposed, by introducing translation, rotation and deformation resistance tensors, which capture each of the flow patterns individually. A parametric study is conducted, considering a wide range of non-spherical particles, determined by the parametric superellipsoid surface equation. Using the results of the parametric study, an approximation scheme is derived on the basis of a multivariate polynomial expression. A coefficient matrix for the polynomial model is introduced, which is used as a tunable parameter for a minimization problem, whereby the polynomials are fitted to the data. The developed model is then put to the test by considering a few examples of particles with different shapes, while also being compared to other, currently available solutions. On top of that, the full functionality of the model is demonstrated by considering an example of a pollen grain, as a realistic non-spherical particle. First, a superellipsoid, which best fits the actual particle shape, is found from the considered range. After that, the coefficients of the translation, rotation and deformation resistance tensors are obtained from the present model and compared to the results of other available models. In the conclusion, a superior accuracy of the present model, for the considered range of particles, is established. To the best of the authors knowledge, this is also one of the first models to extend the torque prediction capabilities beyond sphere and prolate particles. At the same time, the model was demonstrated to be simple to implement and very conservative with the computational resources. As such, it is suitable for large scale studies of dispersed two-phase flows, with a large number of particles.

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Section: Boundary Element Solution Of Stokes Flow Over a Particlementioning

confidence: 99%

Section: Boundary Element Solution Of Stokes Flow Over a Particlementioning

confidence: 99%

A Model for Translation and Rotation Resistance Tensors for Superellipsoidal Particles in Stokes Flow

Štrakl

Hriberšek

Wedel

et al. 2022

JMSE

Self Cite

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“…The numerical implementation of the Stokes BEM solver described in the previous section is based on our Laplace BEM solver [11], [12]. The Stokes solver source code is freely available online.…”

Section: Numerical Implementationmentioning

confidence: 99%

Stokes Flow Induced Drag and Torque on Asbestos-Like Fibres Cannot Be Estimated by a Simplistic Ellipsoidal Approximation

Ravnik

Štrakl

Wedel

et al. 2022

WIT Transactions on Engineering Sciences

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In this paper we present an algorithm based on the Boundary Element Method for solving the Stokes flow around an arbitrarily shaped particle in multiphase flows. The objective of the algorithm is to determine the exact drag force and torque experienced by the particle during its motion in a turbulent flow field. The developed method is used to compare the predicted force and torque values on a realistic 3D model of a mineral fibre and its simplifying ellipsoidal approximation. We found that the force and torque values calculated for the real particle geometry differ greatly from the values obtained with the simplifying ellipsoidal approximation. On average, a relative error of less than 10% is achieved for only 10%-30% of the particle orientations.

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“…, σ n ). In this case, statistics for our deterministic model such as expected value µ y and variance var y may be calculated using [9]:…”

Section: Numerical Code Verificationmentioning

confidence: 99%

“…Evaluating the integral (9) using a standard approach such as Gauss-Lengendre quadrature is possible, however it is very CPU intensive, as the number of evaluations of the model y scales as N n , where N is the number of quadrature sample points. To avoid this, we use the Smolyak [10]- [12] sparse grid approach to numerically evaluate the integral (9). The integral is approximated by…”

Section: Numerical Code Verificationmentioning

confidence: 99%

Coupled Boundary Element: Stochastic Collocation Approach for the Uncertainty Estimation of Simulations of Transcranial Electric Stimulation

Ravnik

Šušnjara

Verhnjak

et al. 2021

WIT Transactions on Engineering Sciences

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In this paper, the authors present a deterministic model of transcranial electric stimulation and develop a Boundary Element Method based algorithm capable of calculating potential and current density in the investigated domain. Furthermore, the deterministic model is coupled with Stochastic Collocation Method to evaluate the propagation of the uncertainty of the input parameters to the results. The uncertainty of the results is analysed via statistical approaches. The governing partial differential equation of the deterministic model is the Laplace equation. We assume that the studied domain is composed of subdomains of different materials. The subdomains are homogeneous and isotropic and have constant electrical conductivity. We introduce the Boundary Element Method for such a setup, support Dirichlet and Neumann boundary conditions at the outer boundary of the domain, and assume continuity of the potential and conservation of current density at the boundaries between the subdomains. We assume that the electrical conductivity of each subdomain is subject to some uncertainty. By coupling the developed simulation tool with the Stochastic Collocation Method, we are able to analyse the uncertainty of the resulting electric potential and current density and identify the contribution to the uncertainty from each subdomain. We apply the developed algorithms to study the transcranial electric stimulation of a human head. A head model with 9 tissues (white and grey matter parts of cerebellum, ventricles, cerebellum, cerebrospinal fluid, head, tongue, cerebrum, and skull) is considered and voltage is applied across two electrodes. The results show that the uncertainty of electrical conductivity in the skull, cerebrospinal fluid and, grey matter have the largest influence on the results.

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Stochastic-deterministic boundary element modelling of transcranial electric stimulation using a three layer head model

Cited by 10 publications

References 35 publications

A Model for Translation and Rotation Resistance Tensors for Superellipsoidal Particles in Stokes Flow

A Model for Translation and Rotation Resistance Tensors for Superellipsoidal Particles in Stokes Flow

Stokes Flow Induced Drag and Torque on Asbestos-Like Fibres Cannot Be Estimated by a Simplistic Ellipsoidal Approximation

Coupled Boundary Element: Stochastic Collocation Approach for the Uncertainty Estimation of Simulations of Transcranial Electric Stimulation

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