A finite element method for viscous membranes

Tasso, Italo Valença Mariotti; Buscaglia, Gustavo C.

doi:10.1016/j.cma.2012.10.021

Cited by 12 publications

(9 citation statements)

References 48 publications

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“…The mathematical formulation of surface viscous behavior was first derived by Scriven [36]. Schemes for its numerical approximation have been proposed by Arroyo and coworkers [4,5,32] in the axisymmetric case, and by Tasso & Buscaglia [40] in the general 3D case. The formulation of this latter article relies heavily on the numerical differentiation of the energy of the membrane (including an "evanescent elasticity" term which accounts for tangential viscosity) to compute forces and stresses, and on yet another numerical differentiation to compute the approximate tangent matrix.…”

Section: Introductionmentioning

confidence: 99%

A semi-implicit finite element method for viscous lipid membranes

Rodrigues

Ausas

Mut

et al. 2015

Journal of Computational Physics

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A finite element formulation to approximate the behavior of lipid membranes is proposed. The mathematical model incorporates tangential viscous stresses and bending elastic forces, together with the inextensibility constraint and the enclosed volume constraint. The membrane is discretized by a surface mesh made up of planar triangles, over which a mixed formulation (velocity-curvature) is built based on the viscous bilinear form (Boussinesq-Scriven operator) and the Laplace-Beltrami identity relating position and curvature. A semi-implicit approach is then used to discretize in time, with piecewise linear interpolants for all variables. Two stabilization terms are needed: The first one stabilizes the inextensibility constraint by a pressure-gradient-projection scheme (R. Codina and J. Blasco, Computer Methods in Applied Mechanics and Engineering 143:373-391, 1997), the second couples curvature and velocity to improve temporal stability, as proposed by Bänsch (Numerische Mathematik 88:203-235, 2001). The volume constraint is handled by a Lagrange multiplier (which turns out to be the internal pressure), and an analogous strategy is used to filter out rigid-body motions. The nodal positions are updated in a Lagrangian manner according to the velocity solution at each time step. An automatic remeshing strategy maintains suitable refinement and mesh quality throughout the simulation.Numerical experiments show the convergent and robust behavior of the proposed method. Stability limits are obtained from numerous relaxation tests, and convergence with mesh refinement is confirmed both in the relaxation transient and in the final equilibrium shape. Virtual tweezing experiments are also reported, computing the dependence of the deformed membrane shape with the tweezing velocity (a purely dynamical effect). For sufficiently high velocities, a tether develops which shows good agreement, both in its final radius and in its transient behavior, with available analytical solutions. Finally, simulation results of a membrane subject to the simultaneous action of six tweezers illustrate the robustness of the method.

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Section: Introductionmentioning

confidence: 99%

A semi-implicit finite element method for viscous lipid membranes

Rodrigues

Ausas

Mut

et al. 2015

Journal of Computational Physics

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“…Hong et al [ 32 ] reported an in-depth study which applied membrane theory to the analysis of circular piezoelectric diaphragm vibration. Please note that this study treated the thin-film PZT and elastic passive layers as thin plates rather than membranes because in finite element studies on membranes, the thickness of membranes tends to be either not considered in meshing or meshed with only a single layer [ 33 , 34 , 35 , 36 , 37 , 38 ]. Kim [ 21 ] conducted model reliability and mesh sensitivity studies of a thin-film, PZT-based structure with different mesh conditions and simulation durations, and concluded that at least three mesh layers in the thin film PZT layer thickness were required to recognize not only motions in the radial or width directions but also those in the thickness direction.…”

Section: Unit-cell Designmentioning

confidence: 99%

10 MHz Thin-Film PZT-Based Flexible PMUT Array: Finite Element Design and Characterization

Kim

Liu

Jackson

et al. 2020

Sensors

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Piezoelectric micromachined ultrasound transducers (PMUT) incorporating lead zirconate titanate PbZr0.52Ti0.48O3 (PZT) thin films were investigated for miniaturized high-frequency ultrasound systems. A recently developed process to remove a PMUT from an underlying silicon (Si) substrate has enabled curved arrays to be readily formed. This research aimed to improve the design of flexible PMUT arrays using PZFlex, a finite element method software package. A 10 MHz PMUT 2D array working in 3-1 mode was designed. A circular unit-cell was structured from the top, with concentric layers of platinum (Pt)/PZT/Pt/titanium (Ti) on a polyimide (PI) substrate. Pulse-echo and spectral response analyses predicted a center frequency of 10 MHz and bandwidth of 87% under water load and air backing. A 2D array, consisting of the 256 (16 × 16) unit-cells, was created and characterized in terms of pulse-echo and spectral responses, surface displacement profiles, crosstalk, and beam profiles. The 2D array showed: decreased bandwidth due to protracted oscillation decay and guided wave effects; mechanical focal length at 2.9 mm; 3.7 mm depth of field for -6 dB; and -55.6 dB crosstalk. Finite element-based virtual prototyping identified figures of merit—center frequency, bandwidth, depth of field, and crosstalk—that could be optimized to design robust, flexible PMUT arrays.

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“…The situation is alleviated to some extent when viscosity of the fluid membrane is considered [22][23][24]. The dependence of the bending energy on membrane curvatures introduces additional challenges in designing numerical schemes.…”

Section: Introductionmentioning

confidence: 99%