# On-two field nurbs based isogeometric formulation for incompressible media problems

## This paper presents u-p mixed formulation relied on the framework of NURBS-based Isogeometric approach (IgA) for incompressible problems. In mixed method, displacement (velocity) field is approximated using NURBS basis functions with one order higher than that of pressure one. | Volume 35 Number 3 3 Vietnam Journal of Mechanics, VAST, Vol. 35, No. 3 (2013), pp. 225 – 237 ON TWO-FIELD NURBS-BASED ISOGEOMETRIC FORMULATION FOR INCOMPRESSIBLE MEDIA PROBLEMS Tran Vinh Loc1 , Thai Hoang Chien1 , Nguyen Xuan Hung1,2,∗ 1 Ton Duc Thang University, Ho Chi Minh City, Vietnam 2 University of Science, VNU-HCMC, Ho Chi Minh City, Vietnam ∗ E-mail: nxhung@ Abstract. This paper presents u-p mixed formulation relied on the framework of NURBS-based Isogeometric approach (IgA) for incompressible problems. In mixed method, displacement (velocity) field is approximated using NURBS basis functions with one order higher than that of pressure one. Being different from the standard FEM, the IgA allows to increase (or decrease) easily the order and continuous derivative of interpolated functions. As a result, a family of NURBS elements, which satisfies the inf-sup condition, is obtained. Benchmark examples are given to validate the excellent performance of the method. Keywords: NURBS, isogeometric, inf-sup, volumetric locking, mixed formulation. 1. INTRODUCTION In computational mechanics, almost of materials are characterized by Young’s modulus E and Poisson’s ratio ν. While Young’s modulus is a measure of the stiffness of an elastic material, Poisson’s ratio ν is defined as the ratio of the lateral compression to the expansion. Mathematically, when ν equals the bulk modulus λ is infinitive, so the system of equilibrium equation becomes highly ill-condition and therefore the accuracy of solution is lost when using lower order finite elements. This phenomenon is called volumetric locking and materials which have ν ≈ are called incompressible materials. Some examples of incompressible or nearly incompressible materials are rubber elasticity, metal plasticity, incompressible flow, etc. To overcome volumetric locking problem, numerous studies have been devised, for example, mixed formulation [1, 2], enhanced assumed strain (EAS) modes [3], .

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