arXiv subjects

Roger A. Sauer

Publications and source records attributed to Roger A. Sauer.

54 records · Page 3Linked to original sources

arXiv2019

An isogeometric finite element formulation for phase transitions on deforming surfaces

This paper presents a general theory and isogeometric finite element implementation for studying mass conserving phase transitions on deforming surfaces. The mathematical problem is governed by two coupled fourth-order nonlinear partial differential equations (PDEs) that live on an evolving two-dimensional manifold. For the phase transitions, the PDE is the Cahn-Hilliard equation for curved surfaces, which can be derived from surface mass balance in the framework of irreversible thermodynamics. For the surface deformation, the PDE is the (vector-valued) Kirchhoff-Love thin shell equation. Both PDEs can be efficiently discretized using $C^1$-continuous interpolations without derivative degrees-of-freedom (dofs). Structured NURBS and unstructured spline spaces with pointwise $C^1$-continuity are utilized for these interpolations. The resulting finite element formulation is discretized in time by the generalized-$α$ scheme with adaptive time-stepping, and it is fully linearized within a monolithic Newton-Raphson approach. A curvilinear surface parameterization is used throughout the formulation to admit general surface shapes and deformations. The behavior of the coupled system is illustrated by several numerical examples exhibiting phase transitions on deforming spheres, tori and double-tori.

math.NA↗

arXiv2019

A concise frictional contact formulation based on surface potentials and isogeometric discretization

This work presents a concise theoretical and computational framework for the finite element formulation of frictional contact problems with arbitrarily large deformation and sliding. The aim of this work is to extend the contact theory based on surface potentials (Sauer and De Lorenzis, 2013) to account for friction. Coulomb friction under isothermal conditions is considered here. For a consistent friction formulation, we start with the first and second laws of thermodynamics and derive the governing equations at the contact interface. A so-called interacting gap can then be defined as a kinematic variable unifying both sliding/sticking and normal/tangential contact. A variational principle for the frictional system can then be formulated based on a purely kinematical constraint. The direct elimination approach applied to the tangential part of this constraint leads to the so-called moving friction cone approach of Wriggers and Haraldsson (2003). Compared with existing friction formulations, our approach reduces the theoretical and computational complexity. Several numerical examples are presented to demonstrate the accuracy and robustness of the proposed friction formulation.

cs.CE↗

arXiv2019

A finite membrane element formulation for surfactants

Surfactants play an important role in various physiological and biomechanical applications. An example is the respiratory system, where pulmonary surfactants facilitate the breathing and reduce the possibility of airway blocking by lowering the surface tension when the lung volume decreases during exhalation. This function is due to the dynamic surface tension of pulmonary surfactants, which depends on the concentration of surfactants spread on the liquid layer lining the interior surface of the airways and alveoli. Here, a finite membrane element formulation for liquids is introduced that allows for the dynamics of concentration-dependent surface tension, as is the particular case for pulmonary surfactants. A straightforward approach is suggested to model the contact line between liquid drops/menisci and planar solid substrates, which allows the presented framework to be easily used for drop shape analysis. It is further shown how line tension can be taken into account. Following an isogeometric approach, NURBS-based finite elements are used for the discretization of the membrane surface. The capabilities of the presented computational model is demonstrated by different numerical examples - such as the simulation of liquid films, constrained and unconstrained sessile drops, pendant drops and liquid bridges - and the results are compared with experimental data.

cond-mat.soft↗

arXiv2019

A nonlinear hyperelasticity model for single layer blue phosphorus based on ab-initio calculations

A new hyperelastic membrane material model is proposed for single layer blue phosphorus ($β\text{-P}$), also known as blue phosphorene. The model is fully nonlinear and captures the anisotropy of $β\text{-P}$ at large strains. The material model is calibrated from density functional theory (DFT) calculations considering a set of elementary deformation states. Those are pure dilatation and uniaxial stretching along the armchair and zigzag directions. The material model is compared and validated with additional DFT results and existing DFT results from the literature, and the comparison shows good agreement. The new material model can be directly used within computational shell formulations that are for example based on rotation-free isogeometric finite elements. This is demonstrated by simulations of the indentation and vibration of single layer blue phosphorus sheets at micrometer scales. The elasticity constants at small deformations are also reported.

cond-mat.mtrl-sci↗

arXiv2019

A nonlinear thermomechanical formulation for anisotropic volume and surface continua

A thermomechanical, polar continuum formulation under finite strains is proposed for anisotropic materials using a multiplicative decomposition of the deformation gradient. First, the kinematics and conservation laws for three dimensional, polar and non-polar continua are obtained. Next, these kinematics are connected to their corresponding counterparts for surface continua based on Kirchhoff-Love kinematics. Likewise, the conservation laws for Kirchhoff-Love shells are derived from their three dimensional counterparts. From this, the weak forms are obtained for three dimensional non-polar continua and Kirchhoff-Love shells. These formulations are expressed in tensorial form so that they can be used in both curvilinear and Cartesian coordinates. They can be used to model anisotropic crystals and soft biological materials, and they can be extended to other field equations, like Maxwell's equations to model thermo-electro-magneto-mechanical materials.

math.NA↗

arXiv2018

A monolithic fluid-structure interaction formulation for solid and liquid membranes including free-surface contact

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 exhibiting strong coupling between fluid and structure. The examples 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.

physics.comp-ph↗

arXiv2018

Computational synthesis of large deformation compliant mechanisms undergoing self and mutual contact

Topologies of large deformation Contact-aided Compliant Mechanisms (CCMs), with self and mutual contact, exemplified via path generation applications, are designed using the continuum synthesis approach. Design domains are parameterized using honeycomb tessellation. Assignment of material to each cell, and generation of rigid contact surfaces, are accomplished via suitably sizing and positioning negative circular masks. To facilitate contact analysis, boundary smoothing is implemented. Mean value coordinates are employed to compute shape functions, as many regular hexagonal cells get degenerated into irregular, concave polygons as a consequence of boundary smoothing. Both, geometric and material nonlinearities are considered in the finite element analysis. The augmented Lagrange multiplier method in association with an active set strategy is employed to incorporate both self and mutual contact. CCMs are evolved using the stochastic hill climber search. Synthesized contact-aided compliant continua trace paths with single and importantly, multiple kinks and experience multiple contact interactions pertaining to both self and mutual contact modes.

cs.CE↗

arXiv2018

A segmentation-free isogeometric extended mortar contact method

This paper presents a new isogeometric mortar contact formulation based on an extended finite element interpolation to capture physical pressure discontinuities at the contact boundary. The so called two-half-pass algorithm is employed, which leads to an unbiased formulation and, when applied to the mortar setting, has the additional advantage that the mortar coupling term is no longer present in the contact forces. As a result, the computationally expensive segmentation at overlapping master-slave element boundaries, usually required in mortar methods (although often simplified with loss of accuracy), is not needed from the outset. For the numerical integration of general contact problems, the so-called refined boundary quadrature is employed, which is based on adaptive partitioning of contact elements along the contact boundary. The contact patch test shows that the proposed formulation passes the test without using either segmentation or refined boundary quadrature. Several numerical examples are presented to demonstrate the robustness and accuracy of the proposed formulation.

cs.CE↗

arXiv2018

A new efficient hyperelastic finite element model for graphene and its application to carbon nanotubes and nanocones

A new hyperelastic material model is proposed for graphene-based structures, such as graphene, carbon nanotubes (CNTs) and carbon nanocones (CNC). The proposed model is based on a set of invariants obtained from the right surface Cauchy-Green strain tensor and a structural tensor. The model is fully nonlinear and can simulate buckling and postbuckling behavior. It is calibrated from existing quantum data. It is implemented within a rotation-free isogeometric shell formulation. The speedup of the model is 1.5 relative to the finite element model of Ghaffari et al. [1], which is based on the logarithmic strain formulation of Kumar and Parks [2]. The material behavior is verified by testing uniaxial tension and pure shear. The performance of the material model is illustrated by several numerical examples. The examples include bending, twisting, and wall contact of CNTs and CNCs. The wall contact is modeled with a coarse grained contact model based on the Lennard-Jones potential. The buckling and post-buckling behavior is captured in the examples. The results are compared with reference results from the literature and there is good agreement.

cs.CE↗

arXiv2018

Modal analysis of graphene-based structures for large deformations, contact and material nonlinearities

The nonlinear frequencies of pre-stressed graphene-based structures, such as flat graphene sheets and carbon nanotubes, are calculated. These structures are modeled with a nonlinear hyperelastic shell model. The model is calibrated with quantum mechanics data and is valid for high strains. Analytical solutions of the natural frequencies of various plates are obtained for the Canham bending model by assuming infinitesimal strains. These solutions are used for the verification of the numerical results. The performance of the model is illustrated by means of several examples. Modal analysis is performed for square plates under pure dilatation or uniaxial stretch, circular plates under pure dilatation or under the effects of an adhesive substrate, and carbon nanotubes under uniaxial compression or stretch. The adhesive substrate is modeled with van der Waals interaction (based on the Lennard-Jones potential) and a coarse grained contact model. It is shown that the analytical natural frequencies underestimate the real ones, and this should be considered in the design of devices based on graphene structures.

cs.CE↗

arXiv2018

The irreversible thermodynamics of curved lipid membranes

The theory of irreversible thermodynamics for arbitrarily curved lipid membranes is presented here. The coupling between elastic bending and irreversible processes such as intra-membrane lipid flow, intra-membrane phase transitions, and protein binding and diffusion is studied. The forms of the entropy production for the irreversible processes are obtained, and the corresponding thermodynamic forces and fluxes are identified. Employing the linear irreversible thermodynamic framework, the governing equations of motion along with appropriate boundary conditions are provided.

cond-mat.soft↗

arXiv2017

A new shell formulation for graphene structures based on existing ab-initio data

An existing hyperelastic membrane model for graphene calibrated from ab-initio data (Kumar and Parks, 2014) is adapted to curvilinear coordinates and extended to a rotation-free shell formulation based on isogeometric finite elements. Therefore, the membrane model is extended by a hyperelastic bending model that reflects the ab-inito data of Kudin et al. (2001). The proposed formulation can be implemented straight-forwardly into an existing finite element package, since it does not require the description of molecular interactions. It thus circumvents the use of interatomic potentials that tend to be less accurate than ab-initio data. The proposed shell formulation is verified and analyzed by a set of simple test cases. The results are in agreement to analytical solutions and satisfy the FE patch test. The performance of the shell formulation for graphene structures is illustrated by several numerical examples. The considered examples are indentation and peeling of graphene and torsion, bending and axial stretch of carbon nanotubes. Adhesive substrates are modeled by the Lennard-Jones potential and a coarse grained contact model. In principle, the proposed formulation can be extended to other 2D materials.

physics.comp-ph↗

arXiv2017

A strategy to interface isogeometric analysis with Lagrangian finite elements - application to fluid-structure interaction problems

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.

physics.flu-dyn↗

arXiv2017

Efficient isogeometric thin shell formulations for soft biological materials

This paper presents three different constitutive approaches to model thin rotation-free shells based on the Kirchhoff-Love hypothesis. One approach is based on numerical integration through the shell thickness while the other two approaches do not need any numerical integration and so they are computationally more efficient. The formulation is designed for large deformations and allows for geometrical and material nonlinearities, which makes it very suitable for the modeling of soft tissues. Furthermore, six different isotropic and anisotropic material models, which are commonly used to model soft biological materials, are examined for the three proposed constitutive approaches. Following an isogeometric approach, NURBS-based finite elements are used for the discretization of the shell surface. Several numerical examples are investigated to demonstrate the capabilities of the formulation. Those include the contact simulation during balloon angioplasty.

cs.CE↗

arXiv2017

Adaptive local surface refinement based on LR NURBS and its application to contact

A novel adaptive local surface refinement technique based on Locally Refined Non-Uniform Rational B-Splines (LR NURBS) is presented. LR NURBS can model complex geometries exactly and are the rational extension of LR B-splines. The local representation of the parameter space overcomes the drawback of non-existent local refinement in standard NURBS-based isogeometric analysis. For a convenient embedding into general finite element code, the Bézier extraction operator for LR NURBS is formulated. An automatic remeshing technique is presented that allows adaptive local refinement and coarsening of LR NURBS. In this work, LR NURBS are applied to contact computations of 3D solids and membranes. For solids, LR NURBS-enriched finite elements are used to discretize the contact surfaces with LR NURBS finite elements, while the rest of the body is discretized by linear Lagrange finite elements. For membranes, the entire surface is discretized by LR NURBS. Various numerical examples are shown, and they demonstrate the benefit of using LR NURBS: Compared to uniform refinement, LR NURBS can achieve high accuracy at lower computational cost.

cs.CE↗

arXiv2016

C1-continuous space-time discretization based on Hamilton's law of varying action

We develop a class of C1-continuous time integration methods that are applicable to conservative problems in elastodynamics. These methods are based on Hamilton's law of varying action. From the action of the continuous system we derive a spatially and temporally weak form of the governing equilibrium equations. This expression is first discretized in space, considering standard finite elements. The resulting system is then discretized in time, approximating the displacement by piecewise cubic Hermite shape functions. Within the time domain we thus achieve C1-continuity for the displacement field and C0-continuity for the velocity field. From the discrete virtual action we finally construct a class of one-step schemes. These methods are examined both analytically and numerically. Here, we study both linear and nonlinear systems as well as inherently continuous and discrete structures. In the numerical examples we focus on one-dimensional applications. The provided theory, however, is general and valid also for problems in 2D or 3D. We show that the most favorable candidate -- denoted as p2-scheme -- converges with order four. Thus, especially if high accuracy of the numerical solution is required, this scheme can be more efficient than methods of lower order. It further exhibits, for linear simple problems, properties similar to variational integrators, such as symplecticity. While it remains to be investigated whether symplecticity holds for arbitrary systems, all our numerical results show an excellent long-term energy behavior.

math.NA↗

arXiv2016

A stabilized finite element formulation for liquid shells and its application to lipid bilayers

This paper presents a new finite element (FE) formulation for liquid shells that is based on an explicit, 3D surface discretization using $C^1$-continuous finite elements constructed from NURBS interpolation. Both displacement-based and mixed FE formulations are proposed. The latter is needed for area-incompressible material behavior, where penalty-type regularizations can lead to misleading results. In order to obtain quasi-static solutions, several numerical stabilization schemes are proposed based on either stiffness, viscosity or projection. Several numerical examples are considered in order to illustrate the accuracy and the capabilities of the proposed formulation, and to compare the different stabilization schemes. The presented formulation is capable of simulating non-trivial surface shapes associated with tube formation and protein-induced budding of lipid bilayers. In the latter case, the presented formulation yields non-axisymmetric solutions, which have not been observed in previous simulations. It is shown that those non-axisymmetric shapes are preferred over axisymmetric ones.

cs.CE↗

arXiv2012

A computational formulation for constrained solid and liquid membranes considering isogeometric finite elements

A geometrically exact membrane formulation is presented that is based on curvilinear coordinates and isogeometric finite elements, and is suitable for both solid and liquid membranes. The curvilinear coordinate system is used to describe both the theory and the finite element equations of the membrane. In the latter case this avoids the use of local cartesian coordinates at the element level. Consequently, no transformation of derivatives is required. The formulation considers a split of the in-plane and out-of-plane membrane contributions, which allows the construction of a stable formulation for liquid membranes with constant surface tension. The proposed membrane formulation is general, and accounts for dead and live loading, as well as enclosed volume, area, and contact constraints. The new formulation is illustrated by several challenging examples, considering linear and quadratic Lagrange elements, as well as isogeometric elements based on quadratic NURBS and cubic T-splines. It is seen that the isogeometric elements are much more accurate than standard Lagrange elements. The gain is especially large for the liquid membrane formulation since it depends explicitly on the surface curvature.

cs.CE↗