volume methods for the integrated modeling of turbulence in the melt and heat transfer ... of the hybrid method. ... ing demands in the performance and producing of Czochralski .... tion sets the free melt surface and melt/crystal interface as pla-.
Mar 27, 2012 - ... not be made otherwise available on the internet; instead link directly to this url for this article. ...... Australia. mailto:[email protected].
Oct 7, 2016 - No-name manuscript No. ... subject to linear boundary conditions on an extended domain, which includes the ... Although these approaches have produced accurate approximations to the EM responses at an affordable.
Jun 11, 2018 - U represents either a scalar or vector quantity whereas A represents a ...... represent the material outward nodal unit normal vector and material ...
Mar 27, 2012 - In the engineering and science fields, the transport equations that ... tries tend to be complex, numerical methods are usually required to solve ... solution variables averaged over their corresponding mesh element (either a ..... cod
Delft University of Technology. Faculty of Electrical Engineering, Mathematics, and Computer Science ... MASTER OF SCIENCE in. APPLIED MATHEMATICS by.
In this paper, we shall construct a coupling method of FVM and the hybrid stress ... We shall overcome these difficulties by a perturbation of B toËB using the ...
Ivan Lunati · Patrick Jenny. Received: 30 March ... e-mail: [email protected]. P. Jenny. Institute of Fluid Dynamics, ETH, Zurich, Switzerland. MSFV method, the ...
Finite volume solution to high rate wave propagation through a lung alveoli stack. ... A finite volume procedure to solve elastic solid mechanics problems in three ...
In our implementation q is the medicenter of the tetrahedron, xijk is the medi- ... one element to control volume Vi in 3-D; Point q is the element's medicenter and ...
A new stable unstructured finite volume method is presented for parallel ... time integration of the incompressible viscoelastic fluid flow equations is an active ...
counterexample to show that an expected L2-error estimate does not exist in ... proved the uniform ellipticity when the dual is Voronoi mesh and each triangle in ... a corresponding covolume Vi as follows: choose any interior point or median, zK, of
large, open networks of mixed finite- and infinite-buffer queues with ... stricted, product-form results, such as those for Jackson networks, can be used.
can be solved efficiently using matrix-geometric or spectral expansion ... exactâ analytic/numerical approach for analysing large open networks of finite-.
analytical and numerical solutions of the two-dimensional pure-convection, convection-diffusion and ... dominated diffusion, and convection-diffusion-react-.
The fast multipole method is used in many scientific computing applications such as .... We have developed a prototype of the FMM algorithm in a Python code, ...
known black box scanners. KeywordsâAutomated vulnerability detection; black-box scanners; web vulnerabilities crawling; security scanner. I. INTRODUCTION.
and its Applications to Speech and Language. Michael ... APIs that make it easy to embed and extend; and (d) is a platfo
Abstract: We give a discrete H1 error estimate of a finite volume ... equation by
various finite volume schemes on regular rectangular structured locally.
boundary conditions model the heat transfer by transepidermal water loss,
radiation, ... Key words. premature infant, bio-heat-transfer-equation, finite
volume ...
(AMR F). For every example we tried to get about the same amount of cells. Initial geometry of first example, as shown on Fig.1, thin square plate. The length.
An unstructured finite volume algorithm for predicting man-made flood routing in ... scheme and a predictor-corrector time stepping procedure which can give the ...
Higher order accuracy in space and time is achieved through a MUSCL-type ..... extended Boussinesq-type equations on unstructured triangular meshes.
Structured quadrilateral meshes. 3. ... in 2D on an structured quadrilateral mesh using the finite volume method .... Integrating on the Control Volume ABCD.
Dec 6, 2013 - for Computational Solid Mechanics ! ! ! School of Mechanical & Materials Engineering. University College Dublin. Joint Symposium of.
5th & 6th December 2013
Joint Symposium of Irish Mechanics Society Irish Society for Scientific & Engineering Computation
An Open Source Finite Volume Method for Computational Solid Mechanics Philip Cardiff A Karac, A Ivankovic, Z Tukovic
School of Mechanical & Materials Engineering University College Dublin ! ! ! !
1
Historical Background & Motivation
2
Finite volume developed for CFD highly nonlinear strong conservation required Multi-physics e.g. Fluid Structure Interaction (FSI)
I. Demirdzic, D. Martinovic, A. Ivankovic, Numerical simulation of thermal deformation in welded workpiece (in croatian), Zavarivanje 31 (1988) 209– 219. 3
2
I
n·Dd
@ u ⇢ 2 =r· @t
Z
Q d⌦ = 0 ⌦
@T 2 ⇢C = kr T @t
Mathematical Model
4
Governing Equations Strong Form of Governing Equation
@ @t
Z
@u ⇢ d⌦ = ⌦ @t
Z
⌦
r·
Momentum
d⌦ +
Z
⌦
⇢f b d⌦
5
Weak Form of Governing Equation
@ @t
Z ✓ ⌦
@u ⇢ @t
◆
· w d⌦ =
Z
⌦
Momentum
· rw d⌦ +
Z
⇢f b · w d⌦ ⌦ Z + t·w d t
6
Weak Form of Governing Equation
@ @t
Z ✓ ⌦
@u ⇢ @t
◆
· w d⌦ =
Z
⌦
Momentum
· rw d⌦ +
weighting function
Z
⇢f b · w d⌦ ⌦ Z + t·w d t
7
2
Strong form is a function of
Weak form is a function of
@ u 2 @x
@u @x
Numerically nicer smaller computational molecule
8
Constitutive Relation Use constitutive law of choice e.g. hypo/hyper elastic, incompressible, etc.
= 2µ✏ + tr(✏)I
1 T ✏= ru + ru 2
Isotropic Hooke’s law
for small strains
9
Development of the Mathematical Model Insert constitutive relation into governing equation
@u ⇢ d⌦ = ⌦ @t
Z
⌦
⇥
⇤
r · µru + µruT + tr(ru)I d⌦ +
Z
⌦
⇢f b d⌦
{
@ @t
Z
10
@ @t
Z
@u ⇢ d⌦ = ⌦ @t
Z
µr2 u d⌦ + ⌦
Z
⌦
⇥
⇤
r · µruT + tr(ru)I d⌦ +
Z
⌦
⇢f b d⌦
11
laplacian term implicitly approximated using compact molecule @ @t
Z
@u ⇢ d⌦ = ⌦ @t
Z
µr2 u d⌦ + ⌦
Z
⌦
⇥
⇤
r · µruT + tr(ru)I d⌦ +
Z
⌦
⇢f b d⌦
12
Nonlinear stress analysis Choice of total/updated/deformed Lagrangian approaches Choice of stress tensor (or stress rate) may be used P. Cardiff et al. / Comput. Methods Appl. Mech. Engrg. 268 (2014) 318–335
@ @t
Z
@ðu þ duÞ q dXu ¼ @t Xu
I
nu % ðC u : dEu Þ dCu þ Cu Z þ q ðb þ dbÞ dXu
I
Cu
nu % ½ðS u þ dS u Þ % rdu' dCu þ
I
Cu
nu % S u dCu
Xu
where dF u ¼ rdu, and the increment of Green strain (Eq. (3)) reduces to:
dEu ¼
1!
T
rdu þ rdu þ rdu % rdu
T
"
13
Discrete System outer iterations are performed until
Au = b
b
has converged linear system is sparse
diagonally dominant iteratively solved
14
FEM & FVM Comparison
for solids FVM
Pros
Cons
attractively simple strongly conservative memory efficient straight-forward nonlinearity suitable for FSI/cracks general polyhedral cells
segregated solution not always efficient higher order elements are more difficult less development 15
distinguishing feature of OpenFOAM is that the partial differential equation and tensor operations syntax closely resemthe equations being solved. An extract of the code from the developed elasticOrthoNonLinULSolidFoam solver, ementing the developed orthotropic updated Lagrangian approach, is given in Appendix (A), and shows remarkable simy to the previously described mathematical model.
Accuracy...
Orthotropic plate with a hole
25
T
2
2
4
sin h þ
!k cos h þ ð1 þ nÞ sin h ðn2
2
! 2kÞ sin
h cos2
2
h þ k cos4 h
polyhedral mesh
26
second order accuracy
27
Parallelisation
28
29
Summary & Conclusions
30
Finite volume may not optimal for all problems but it is a viable alternative to FE in many applications
31
Open source may not optimal for all problems but it is a viable alternative in many applications
32
5th & 6th December 2013
Joint Symposium of Irish Mechanics Society Irish Society for Scientific & Engineering Computation
An Open Source Finite Volume Method for Computational Solid Mechanics Philip Cardiff A Karac, A Ivankovic, Z Tukovic
School of Mechanical & Materials Engineering University College Dublin ! ! ! !
