In this paper, we develop a geometrically flexible technique for computational fluid–structure interaction (FSI). The motivating application is the simulation of tri-leaflet bioprosthetic heart valve ...function over the complete cardiac cycle. Due to the complex motion of the heart valve leaflets, the fluid domain undergoes large deformations, including changes of topology. The proposed method directly analyzes a spline-based surface representation of the structure by immersing it into a non-boundary-fitted discretization of the surrounding fluid domain. This places our method within an emerging class of computational techniques that aim to capture geometry on non-boundary-fitted analysis meshes. We introduce the term “immersogeometric analysis” to identify this paradigm.
The framework starts with an augmented Lagrangian formulation for FSI that enforces kinematic constraints with a combination of Lagrange multipliers and penalty forces. For immersed volumetric objects, we formally eliminate the multiplier field by substituting a fluid–structure interface traction, arriving at Nitsche’s method for enforcing Dirichlet boundary conditions on object surfaces. For immersed thin shell structures modeled geometrically as surfaces, the tractions from opposite sides cancel due to the continuity of the background fluid solution space, leaving a penalty method. Application to a bioprosthetic heart valve, where there is a large pressure jump across the leaflets, reveals shortcomings of the penalty approach. To counteract steep pressure gradients through the structure without the conditioning problems that accompany strong penalty forces, we resurrect the Lagrange multiplier field. Further, since the fluid discretization is not tailored to the structure geometry, there is a significant error in the approximation of pressure discontinuities across the shell. This error becomes especially troublesome in residual-based stabilized methods for incompressible flow, leading to problematic compressibility at practical levels of refinement. We modify existing stabilized methods to improve performance.
To evaluate the accuracy of the proposed methods, we test them on benchmark problems and compare the results with those of established boundary-fitted techniques. Finally, we simulate the coupling of the bioprosthetic heart valve and the surrounding blood flow under physiological conditions, demonstrating the effectiveness of the proposed techniques in practical computations.
Left ventricular assist devices (LVADs) are continuous flow pumps that are employed in patients with severe heart failure. Although their emergence has significantly improved therapeutic options for ...patients with heart failure, detailed studies of the impact of LVADs on hemodynamics are notably lacking. To this end we initiate a computational study of the Jarvik 2000 LVAD model employing isogeometric fluid–structure interaction analysis. We focus on a patient-specific configuration in which the LVAD is implanted in the descending thoracic aorta. We perform computations for three pump settings and report our observations for several quantities of hemodynamic interest. It should be noted that this paper presents the first three-dimensional, patient-specific fluid–structure interaction simulation of LVADs.
We investigate the effects of smoothness of basis functions on solution accuracy within the isogeometric analysis framework. We consider two simple one-dimensional structural eigenvalue problems and ...two static shell boundary value problems modeled with trivariate NURBS solids. We also develop a local refinement strategy that we utilize in one of the shell analyses. We find that increased smoothness, that is, the “
k-method,” leads to a significant increase in accuracy for the problems of structural vibrations over the classical
C
0
-continuous “
p-method,” whereas a judicious insertion of
C
0
-continuous surfaces about singularities in a mesh otherwise generated by the
k-method, usually outperforms a mesh in which all basis functions attain their maximum level of smoothness. We conclude that the potential for the
k-method is high, but smoothness is an issue that is not well understood due to the historical dominance of
C
0
-continuous finite elements and therefore further studies are warranted.
The concept of isogeometric analysis is proposed. Basis functions generated from NURBS (Non-Uniform Rational B-Splines) are employed to construct an exact geometric model. For purposes of analysis, ...the basis is refined and/or its order elevated without changing the geometry or its parameterization. Analogues of finite element
h- and
p-refinement schemes are presented and a new, more efficient, higher-order concept,
k-refinement, is introduced. Refinements are easily implemented and exact geometry is maintained at all levels without the necessity of subsequent communication with a CAD (Computer Aided Design) description. In the context of structural mechanics, it is established that the basis functions are complete with respect to affine transformations, meaning that all rigid body motions and constant strain states are exactly represented. Standard patch tests are likewise satisfied. Numerical examples exhibit optimal rates of convergence for linear elasticity problems and convergence to thin elastic shell solutions. A
k-refinement strategy is shown to converge toward monotone solutions for advection–diffusion processes with sharp internal and boundary layers, a very surprising result. It is argued that isogeometric analysis is a viable alternative to standard, polynomial-based, finite element analysis and possesses several advantages.
This paper uses a divergence-conforming B-spline fluid discretization to address the long-standing issue of poor mass conservation in immersed methods for computational fluid–structure interaction ...(FSI) that represent the influence of the structure as a forcing term in the fluid subproblem. We focus, in particular, on the immersogeometric method developed in our earlier work, analyze its convergence for linear model problems, then apply it to FSI analysis of heart valves, using divergence-conforming B-splines to discretize the fluid subproblem. Poor mass conservation can manifest as effective leakage of fluid through thin solid barriers. This leakage disrupts the qualitative behavior of FSI systems such as heart valves, which exist specifically to block flow. Divergence-conforming discretizations can enforce mass conservation exactly, avoiding this problem. To demonstrate the practical utility of immersogeometric FSI analysis with divergence-conforming B-splines, we use the methods described in this paper to construct and evaluate a computational model of an in vitro experiment that pumps water through an artificial valve.
•Div-conforming B-splines improve immersed fluid–structure interaction (FSI) analysis.•Strong mass conservation prevents spurious non-physical leakage through barriers.•Semi-implicit time integration is shown to converge a priori for a model problem.•Div-conforming immersogeometric FSI analysis is practical for heart valve analysis.•FSI simulations reproduce qualitative features of in vitro experiments.
We introduce provably unconditionally stable mixed variational methods for phase-field models. Our formulation is based on a mixed finite element method for space discretization and a new ...second-order accurate time integration algorithm. The fully-discrete formulation inherits the main characteristics of conserved phase dynamics, namely, mass conservation and nonlinear stability with respect to the free energy. We illustrate the theory with the Cahn–Hilliard equation, but our method may be applied to other phase-field models. We also propose an adaptive time-stepping version of the new time integration method. We present some numerical examples that show the accuracy, stability and robustness of the new method.
We present an LES-type variational multiscale theory of turbulence. Our approach derives completely from the incompressible Navier–Stokes equations and does not employ any
ad hoc devices, such as ...eddy viscosities. We tested the formulation on forced homogeneous isotropic turbulence and turbulent channel flows. In the calculations, we employed linear, quadratic and cubic NURBS. A dispersion analysis of simple model problems revealed NURBS elements to be superior to classical finite elements in approximating advective and diffusive processes, which play a significant role in turbulence computations. The numerical results are very good and confirm the viability of the theoretical framework.
Isogeometric analysis of structural vibrations Cottrell, J.A.; Reali, A.; Bazilevs, Y. ...
Computer methods in applied mechanics and engineering,
08/2006, Letnik:
195, Številka:
41
Journal Article
Recenzirano
Odprti dostop
This paper begins with personal recollections of John H. Argyris. The geometrical spirit embodied in Argyris’s work is revived in the sequel in applying the newly developed concept of isogeometric ...analysis to structural vibration problems. After reviewing some fundamentals of isogeometric analysis, application is made to several structural models, including rods, thin beams, membranes, and thin plates. Rotationless beam and plate models are utilized as well as three-dimensional solid models. The concept of
k-refinement is explored and shown to produce more accurate and robust results than corresponding finite elements. Through the use of nonlinear parameterization, “optical” branches of frequency spectra are eliminated for
k-refined meshes. Optical branches have been identified as contributors to Gibbs phenomena in wave propagation problems and the cause of rapid degradation of higher modes in
p-method finite elements. A geometrically exact model of the NASA Aluminum Testbed Cylinder is constructed and frequencies and mode shapes are computed and shown to compare favorably with experimental results.
Following a series of recent innovations, isogeometric shell analysis based on trimmed CAD surfaces is currently being developed into an accurate, efficient and mature design-through-analysis ...methodology. This work contributes to this emerging technology with respect to the following aspects. On the analysis side, we present a robust variationally consistent Nitsche-type formulation for thin shells at large deformations that weakly enforces coupling constraints at trimming curves. On the geometry side, we present a set of algorithms that enable automatic interaction of trimmed shell analysis with CAD data structures based on the STEP exchange format. We integrate these methodologies in a comprehensive framework for isogeometric trimmed shell analysis. We demonstrate that our framework is able to seamlessly perform large-deformation stress analysis of an industry-scale 76-patch surface model of a Dodge RAM hood, while delivering comparable accuracy with respect to Simulia’s commercial software package Abaqus.
•We present a variationally consistent formulation for coupling isogeometric shells at trimming curves.•We also present algorithms that enable automatic interaction with CAD data structures based on the STEP exchange format.•We integrate these methodologies in a comprehensive framework for isogeometric trimmed shell analysis.•It enables seamless and accurate large-deformation stress analysis, illustrated with a 76-patch model of a Dodge RAM hood.
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