Scaled boundary finite element method with exact defining curves for two-dimensional linear multi-field media

doi:10.1007/s11709-018-0488-0

Front. Struct. Civ. Eng.

2019, Vol. 13

Issue (1) : 201-214 https://doi.org/10.1007/s11709-018-0488-0

RESEARCH ARTICLE

Scaled boundary finite element method with exact defining curves for two-dimensional linear multi-field media

Jaroon RUNGAMORNRAT¹, Chung Nguyen VAN^1,²(

)

¹. Applied Mechanics and Structures Research Unit, Department of Civil Engineering, Faculty of Engineering, Chulalongkorn University, Bangkok 10330, Thailand
². Faculty of Civil Engineering, Ho Chi Minh City of Technology and Education, Ho Chi Minh 721400, Vietnam

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Abstract

This paper presents an efficient and accurate numerical technique based upon the scaled boundary finite element method for the analysis of two-dimensional, linear, second-order, boundary value problems with a domain completely described by a circular defining curve. The scaled boundary finite element formulation is established in a general framework allowing single-field and multi-field problems, bounded and unbounded bodies, distributed body source, and general boundary conditions to be treated in a unified fashion. The conventional polar coordinates together with a properly selected scaling center are utilized to achieve the exact description of the circular defining curve, exact geometry of the domain, and exact spatial differential operators. Standard finite element shape functions are employed in the discretization of both trial and test functions in the circumferential direction and the resulting eigenproblem is solved by a selected efficient algorithm. The computational performance of the implemented procedure is then fully investigated for various scenarios to demonstrate the accuracy in comparison with standard linear elements.

Keywords multi-field problems defining curve exact geometry general boundary conditions SBFEM

Corresponding Author(s): Chung Nguyen VAN

Just Accepted Date: 14 May 2018 Online First Date: 27 June 2018 Issue Date: 04 January 2019

Cite this article:

Jaroon RUNGAMORNRAT,Chung Nguyen VAN. Scaled boundary finite element method with exact defining curves for two-dimensional linear multi-field media[J]. Front. Struct. Civ. Eng., 2019, 13(1): 201-214.

URL:

https://academic.hep.com.cn/fsce/EN/10.1007/s11709-018-0488-0
https://academic.hep.com.cn/fsce/EN/Y2019/V13/I1/201

Fig.1 Schematic of two-dimensional, multi-field body subjected to external excitations

Fig.2 Schematic of generic body

Ω

, corresponding boundary, and its approximation

Ω h

Fig.3 Schematic of scaling center and defining curve of (a) 2-node isoparametric linear element and (b) 2-node circular-arc element

Fig.4 Schematic of a quarter of a ring subjected to non-uniform heat source and mixed boundary conditions

n	$u 1 / u 1 exact$
	$θ / π$ =0.1		$θ / π$ =0.2		$θ / π$ =0.3			$θ / π$ =0.4
	Type-1	Type-2	Type-1	Type-2	Type-1	Type-2	Type-1		Type-2
2	1.0609	1.0000	1.1218	1.0000	1.1483	1.0000	1.1405		1.0000
4	1.0252	1.0000	1.0336	1.0000	1.0352	1.0000	1.0354		1.0000
8	1.0067	1.0000	1.0085	1.0000	1.0088	1.0000	1.0089		1.0000
16	1.0017	1.0000	1.0021	1.0000	1.0022	1.0000	1.0022		1.0000

Tab.1 Normalized temperatures

u 1 / u 1 exact

along the circular arc between AD and BC of two-dimensional domain associated with quarter of ring

Fig.5 Relative errors of SBFE solutions versus number of degrees of freedom used in discretization of circular defining curve. DF: Defining curve

Fig.6 Schematics of (a) plane-strain hollowed disk fully restrained against the movement on its outer boundary and subjected to uniform shear traction on its inner boundary and (b) equivalent quarter of body used in analysis

n	$u 1 / u 1 exact$
	$θ / π$ =0.1		$θ / π$ =0.2		$θ / π$ =0.3			$θ / π$ =0.4
	Type-1	Type-2	Type-1	Type-2	Type-1	Type-2	Type-1		Type-2
2	0.7449	0.9529	0.7832	1.0019	0.7702	0.9853	0.7554		0.9664
4	0.9411	1.0009	0.9330	0.9923	0.9320	0.9913	0.9379		0.9975
8	0.9829	0.9981	0.9840	0.9993	0.9843	0.9996	0.9826		0.9979
16	0.9960	0.9998	0.9956	0.9995	0.9956	0.9995	0.9960		0.9999

Tab.2 Normalized displacements

u 1 / u 1 exact

along circular arc between AD and BC of plane-strain hollowed disk

n	$u 2 / u 2 e x a c t$
	$θ / π$ =0.1		$θ / π$ =0.2		$θ / π$ =0.3			$θ / π$ =0.4
	Type-1	Type-2	Type-1	Type-2	Type-1	Type-2	Type-1		Type-2
2	0.7554	0.9664	0.7702	0.9853	0.7832	1.0019	0.7449		0.9529
4	0.9379	0.9975	0.9320	0.9913	0.9330	0.9923	0.9411		1.0009
8	0.9826	0.9979	0.9843	0.9996	0.9840	0.9993	0.9829		0.9981
16	0.9960	0.9999	0.9956	0.9995	0.9956	0.9995	0.9960		0.9998

Tab.3 Normalized displacements

u 2 / u 2 exact

along circular arc between AD and BC of plane-strain hollowed disk

Fig.7 Relative errors of SBFE solutions versus number of degrees of freedom used in discretization of circular defining curve. DF: Defining curve

n	$u 1 / u 1 exact$
	$x ‾ 1$ =1.25		$x ‾ 1$ =1.50		$x ‾ 1$ =1.75		$x ‾ 1$ =2.00
	Type-1	Type-2	Type-1	Type-2	Type-1	Type-2	Type-1	Type-2
2	0.7368	0.8935	0.7228	0.9262	0.7799	1.0043	0.8605	1.1004
4	0.9361	0.9806	0.9309	0.9936	0.9385	1.0115	0.9413	1.0199
8	0.9830	0.9946	0.9812	0.9974	0.9830	1.0014	0.9852	1.0046
16	0.9957	0.9986	0.9952	0.9993	0.9957	1.0003	0.9963	1.0011

Tab.4 Normalized displacements

u 1 / u 1 exact

along the line

x 1 = x 2

of quarter of hollowed circular plate

n	$u 2 / u 2 exact$
	$x ‾ 1$ =1.25		$x ‾ 1$ =1.50		$x ‾ 1$ =1.75		$x ‾ 1$ =2.00
	Type-1	Type-2	Type-1	Type-2	Type-1	Type-2	Type-1	Type-2
2	1.0289	0.9705	1.0421	0.9801	1.0493	0.9917	1.0525	0.9981
4	1.0072	0.9984	1.0088	0.9987	1.0097	0.9998	1.0111	1.0019
8	1.0016	0.9997	1.0021	0.9998	1.0023	0.9999	1.0022	1.0000
16	1.0004	0.9999	1.0005	1.0000	1.0006	1.0000	1.0005	1.0000

Tab.5 Normalized displacements

u 2 / u 2 exact

along the line

x 1 = x 2

of quarter of hollowed circular plate

n	$u 3 / u 3 exact$
	$x ‾ 1$ =1.25		$x ‾ 1$ =1.50		$x ‾ 1$ =1.75		$x ‾ 1$ =2.00
	Type-1	Type-2	Type-1	Type-2	Type-1	Type-2	Type-1	Type-2
2	0.9864	1.0111	0.9699	1.0088	0.9500	0.9992	0.9268	0.9844
4	0.9973	1.0018	0.9944	1.0019	0.9914	1.0014	0.9892	1.0010
8	0.9993	1.0004	0.9986	1.0004	0.9978	1.0003	0.9970	1.0000
16	0.9998	1.0001	0.9996	1.0001	0.9994	1.0001	0.9992	1.0000

Tab.6 Normalized electric potential

u 3 / u 3 exact

along the line

x 1 = x 2

of quarter of hollowed circular plate

Fig.8 Relative errors of SBFE solutions versus number of degrees of freedom used in discretization of circular defining curve. DF: Defining curve

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[1]	Chung Nguyen VAN. Numerical investigation of circle defining curve for two-dimensional problem with general boundaries using the scaled boundary finite element method[J]. Front. Struct. Civ. Eng., 2019, 13(1): 92-102.
[2]	DU Jianguo, LIN Gao. Improved numerical method for time domain dynamic structure-foundation interaction analysis based on scaled boundary finite element method[J]. Front. Struct. Civ. Eng., 2008, 2(4): 336-342.

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