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Beltrami flow: Convergence of the velocity error norm e v with mesh refinement for the Beltrami flow case at t = 10 s.
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Motivated by the superior accuracy and better stability of isogeometrically enriched finite elements - when compared to standard Lagrangian finite elements for problems involving contact and debonding [15,16] - we extend their applicability to fluid flow problems. Internal and external flow involving incompressible Newtonian fluids is analyzed in t...
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... for the mesh convergence study for error norms e v and e p at t = 10 are shown in Figure 3 and Figure 4. Here h denotes the average element length in B h . ...
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Citations
... Since Hughes et al. [17] proposed the IGA method, many researchers have further developed it. Currently, this method has been applied in many fields, such as heat conduction problems [18], fluid flow problems [19,20], acoustic problems [21], electromagnetic problems [22], and fluid structure coupling problems [23][24][25][26][27][28]. IGA has also been applied in the medical domain [29][30]. ...
Gears are pivotal in mechanical drives, and gear contact analysis is a typically difficult problem to solve. Emerging isogeometric analysis (IGA) methods have developed new ideas to solve this problem. In this paper, a three-dimensional body parametric gear model of IGA is established, and a theoretical formula is derived to realize single-tooth contact analysis. Results were benchmarked against those obtained from commercial software utilizing the finite element analysis (FEA) method to validate the accuracy of our approach. Our findings indicate that the IGA-based contact algorithm successfully met the Hertz contact test. When juxtaposed with the FEA approach, the IGA method demonstrated fewer node degrees of freedom and reduced computational units, all while maintaining comparable accuracy. Notably, the IGA method appeared to exhibit consistency in analysis accuracy irrespective of computational unit density, and also significantly mitigated non-physical oscillations in contact stress across the tooth width. This underscores the prowess of IGA in contact analysis. In conclusion, IGA emerges as a potent tool for addressing contact analysis challenges and holds significant promise for 3D gear modeling, simulation, and optimization of various mechanical components.
... Modeling the whole problem domain with NURBS often needs multi-patch approaches (Nguyen et al. 2014b;Kiendl et al. 2010) besides the high computational cost of NURBS elements. Coupling of traditional finite elements with NURBS elements is an interesting solution to exploit computational efficiency of FEs and also circumvent multi-patch approaches (Rasool et al. 2016;Ge et al. 2016;Corbett and Sauer 2014). In this paper the technique proposed in Corbett and Sauer (2014) is utilized to do the coupling task. ...
In this paper, employing a new numerical framework, a 2D investigation is conducted on the effect of fiber-matrix contact/debonding on the stress–strain response of fiber-reinforced elastomeric composites. The analysis is carried out in the framework of large deformation/sliding. The main novelty of this work resides in using hybrid IsoGeometric-Finite Element (IG-FE) technique to discretize the domain where interfaces together with a band around them are represented by NURBS patches. These patches are coupled to the surrounding Lagrangian domain thanks to the transition elements. Interfaces’ discontinuity is readily induced through knot insertion capability within IGA. Moreover, contact constraints together with cohesive behavior are incorporated in the numerical model employing a unified mortar framework with appropriate definitions of potentials. Two numerical examples are considered within the proposed context and results are discussed in terms of stress–strain curves and stress contours. Due to proven higher accuracy of IGA in modeling contact/cohesive interfaces under large deformation and efficient use of NURBS around discontinuities, the proposed numerical methodology seems to be an appropriate (yet efficient) candidate to model such kind of highly nonlinear problems.
... In the context of finite element contact analysis, a considerable research progress has been made that integrates the intrinsic features of NURBS and higher-order Lagrange polynomials with the standard FE discretization [44][45][46][47][48][49]. Corbett and Sauer [44,45] introduced a NURBS-enriched contact element formulation that combines the geometric smoothness of NURBS with the efficiency characteristic of the FE discretization. ...
A novel varying-order based NURBS discretization method is proposed to enhance the performance of isogeometric analysis (IGA) technique within the framework of computational contact mechanics. The method makes use of higher-order NURBS for the contact integral evaluations. The minimum orders of NURBS capable of representing the complex geometries exactly are employed for the bulk computations. The proposed methodology provides a possibility to refine the geometry through controllable order elevation for isogeometric analysis. To achieve this, a higher-order NURBS layer is used as the contact boundary layer of the bodies. The NURBS layer is constructed using different surface refinement strategies such that it is accompanied by a large number of additional degrees of freedom and matches with the bulk parameterization.
The capabilities and benefits of the proposed method are demonstrated using the two-dimensional frictionless and frictional contact problems, considering both small and large deformations. The results with the existing fixed-order based NURBS discretizations are used for comparisons. Numerical examples show that with the proposed method, much higher accuracy can be achieved even with a coarse mesh as compared to the existing NURBS discretization approach. It exhibits a major gain in the numerical efficiency without the loss of stability, robustness, and the intrinsic features of NURBS-based IGA technique for a similar accuracy level. The simplicity of the proposed method lends itself to be conveniently embedded in an existing isogeometric contact code after only a few minor modifications.
... Thus, IGA exhibits better numerical accuracy and efficiency in comparison to FEM. Up to now IGA has been successfully adopted by a large number of researchers in applications as different as fracture problems [31], structure vibration [32], dynamic problems [33], incompressible flow problems [34], contact mechanics [35], optimization problems [36], and so on. ...
... Isogeometrically enriched finite elements , Rasool et al. [2016], Harmel et al. [2017]) offer efficient localized isogeometric analysis (IGA) enrichment for numerical simulations involving large computational domains. This is achieved by employing surface enriched elements to interface isogeometric elements with classical Langrangian finite elements. ...
... From biomedical applications (Zhang et al. [2007], , Chivukula et al. [2014]) to the design of wind turbines (Bazilevs et al. [2011(Bazilevs et al. [ , 2012(Bazilevs et al. [ , 2016), IGA has been successively applied to various FSI problems. The superiority of IGA over interpolatory FEM for uncoupled problems has been investigated by many researchers, e.g. , Akkerman et al. [2008], Motlagh et al. [2013], Rasool et al. [2016] for fluid dynamics and Cottrell et al. [2006], , , Morganti et al. [2015] for structural mechanics applications. For many of these studies, IGA yields higher accuracy per degree-offreedom (dof) than the classical FEM. ...
... For many problems, especially true for simulation of physical flow in unbounded regions, it is sufficient to have an enriched analysis only in certain localized regions of the discretized domain, while the remaining bulk can be represented with a low-order computationally efficient discretization, e.g. Rasool et al. [2016] and Harmel et al. [2017]. Such applications motivates a framework where an interpolatory finite element discretization is enriched with isogeometric elements only at regions where an enhanced representation and analysis is beneficial. ...
Isogeometrically enriched finite elements offer efficient localized isogeometric analysis (IGA) enrichment for numerical simulations involving large computational domains. This is achieved by employing surface enriched elements to interface isogeometric elements with classical Langrangian finite elements. In this paper, we explore their applicability and merits for fluid-structure interaction (FSI) analysis. The implemented approach not only offers an enrichment of the finite element space, but also offers a framework for discretizing and analyzing fluid and structure with different finite element approaches, namely, classical Lagrange finite elements and IGA. In this context, a monolithic solution approach with an explicit grid update mechanism is implemented for FSI. The applicability and the impact of the isogeometric enrichment approach on the accuracy of the numerical solution is assessed by comparing the obtained results with existing reference solutions of FSI benchmark examples involving two- and three-dimensional incompressible fluid flow past hyper-elastic solids.
... The stabilization parameters τ v and τ p appearing inside f e supg and g e pspg are computed from (Shakib, 1988;Tezduyar, 1992;Rasool et al., 2016), where ∆t is the time step size, h e is the "element length" in the local flow direction taken from (Tezduyar, 1992) and m e depends on the polynomical order of the shape functions. I.e. for L1 (linear Lagrange) and L2 (quadratic Lagrange) elements we have m e = 1/3 and m e = 1/12, respectively. ...
... In order to increase efficiency, the formulation can be extended to boundary elements (for low Re) or turbulence models (for high Re). Under current study is the use of enriched finite element discretizations (Harmel et al., 2017) that are suitable to efficiently capture boundary layers (Rasool et al., 2016). Another extension of the present formulation is to re-examine the pressure stabilization scheme at contact boundaries. ...
A unified fluid-structure interaction (FSI) formulation is presented for solid, liquid and mixed membranes. Nonlinear finite elements (FE) and the generalized-alpha scheme are used for the spatial and temporal discretization. The membrane discretization is based on curvilinear surface elements that can describe large deformations and rotations, and also provide a straightforward description for contact. The fluid is described by the incompressible Navier-Stokes equations, and its discretization is based on stabilized Petrov-Galerkin FE. The coupling between fluid and structure uses a conforming sharp interface discretization, and the resulting non-linear FE equations are solved monolithically within the Newton-Raphson scheme. An arbitrary Lagrangian-Eulerian formulation is used for the fluid in order to account for the mesh motion around the structure. The formulation is very general and admits diverse applications that include contact at free surfaces. This is demonstrated by two analytical and three numerical examples. They include balloon inflation, droplet rolling and flapping flags. They span a Reynolds-number range from 0.001 to 2000. One of the examples considers the extension to rotation-free shells using isogeometric FE.
... Sauer (2014, 2015) show good results for contact problems in solid mechanics. The creation of boundary layers (zone enrichment) gives numerical benefits in fluid mechanics (Rasool et al., 2016). The examples considered in these works use structured meshes and simple geometries. ...
... In several applications, a pure isogeometric boundary zone seems promising (see Rasool et al. (2016)). To this end, one single or multiple layers of pure isogeometric elements are placed close to the surface. ...
... Isogeometrically enriched elements can also be used to mesh the surrounding of an object by applying the concept of SEN and ZEN to the outer side of a bodies surface (Rasool et al., 2016). This application of the method can be inter alia useful to create models for flow simulations. ...
A new approach to obtain a volumetric discretization from a T-spline surface representation is presented. A T-spline boundary zone is created beneath the surface, while the core of the model is discretized with Lagrangian elements. T-spline enriched elements are used as an interface between isogeometric and Lagrangian finite elements. The thickness of the T-spline zone and thereby the isogeometric volume fraction can be chosen arbitrarily large such that pure Lagrangian and pure isogeometric discretizations are included. The presented approach combines the advantages of isogeometric elements (accuracy and smoothness) and classical finite elements (simplicity and efficiency).
Different heat transfer problems are solved with the finite element method using the presented discretization approach with different isogeometric volume fractions. For suitable applications, the approach leads to a substantial accuracy gain.
A novel varying-order based NURBS discretization method is proposed to enhance the performance of isogeometric analysis (IGA) technique within the framework of computational contact mechanics. The method makes use of higher-order NURBS for the contact integral evaluations. The minimum orders of NURBS capable of representing the complex geometries exactly are employed for the bulk computations. The proposed methodology provides a possibility to refine the geometry through controllable order elevation for isogeometric analysis. To achieve this, a higher-order NURBS layer is used as the contact boundary layer of the bodies. The NURBS layer is constructed using different surface refinement strategies such that it is accompanied by a large number of additional degrees of freedom and matches with the bulk parameterization.
The capabilities and benefits of the proposed method are demonstrated using the two-dimensional frictionless and frictional contact problems, considering both small and large deformations. The results with the existing fixed-order based NURBS discretizations are used for comparisons. Numerical examples show that with the proposed method, a much higher accuracy can be achieved even with a coarse mesh as compared to the existing NURBS discretization approach. It exhibits a major gain in the numerical efficiency without the loss of stability, robustness, and the intrinsic features of NURBS-based IGA technique for a similar accuracy level. The simplicity of the proposed method lends itself to be conveniently embedded in an existing isogeometric contact code after only a few minor modifications.
Isogeometric analysis (IGA) employs non-uniform rational B-splines (NURBS) or other B-spline-based variants to represent both the geometry and the field variable. Exact geometry representation and higher order global continuity (at least C1 even on elements' boundaries) are two favorable properties that would make IGA an appropriate discretization technique in problems with responses associated with the derivatives of the primary field variable. As a category of these problems, in this paper, 2D elastostatic problems involving stress concentration sites are analyzed with a hybrid isogeometric-finite element (IG-FE) discretization. To exploit higher order continuity of NURBS basis functions, IGA discretization is applied selectively at pre-identified locations of high displacement gradients where the stress concentration occurs. In addition, considering computational efficiency, the rest of problem domain is discretized by means of linear Lagrangian finite elements. The connection of NURBS and Lagrangian domain is carried out through employing specially devised elements [Corbett, C. J. and Sauer, R. A. [2014] "NURBS-enriched contact finite elements", Computer Methods in Applied Mechanics and Engineering 275, 55-75]. The methodology is applied in some 2D elastostatic examples. Increasing the number of DOFs and comparing convergence of the concentrated stress value using different discretizations, it is shown that the hybrid IG-FE discretizations generally have faster and more stable convergence response compared with pure FE discretizations especially at lower DOFs.
A hybrid NURBS-based isogeometric-finite element discretization technique is presented to take advantages of IGA for problems with regions demanding higher geometric accuracy or interpolation order. The validity of this coupled discretization is tested through standard patch test. Also, for demonstrative purposes, the proposed methodology is then applied to 2D interface contact/debonding of fiber–matrix as well as a 2D double cantilever beam peeling problem. Adopting a novel approach, the mortar method is used to account for both cohesive behavior and non-penetration constraint in the interface. Non-penetration constraint is enforced via Lagrange multiplier technique along with a mixed-mode cohesive law. The validity of the proposed unified contact/debonding formulation is tested against a contact/tension patch test. The stress contours after simulations show a smooth variation across IG and FE domains further demonstrating the coupling’s eligibility. Moreover, smooth contact pressure distributions are obtained along the interfaces resulting from higher-order NURBS basis functions. Having significantly less DOFs, the hybrid IG–FE discretization also behaved more robustly compared with linear FEs.
