Research projects
DFG GR 3297/10-1: A structure-preserving immersed finite element method for the dynamics of multiphase continua with thermomechanical coupling.
Concerning finite element simulations of moving continua,
there are many examples in which a considered continuum consists of different embedded phases.
Here, the phases can be flexible solids, fluids as well as rigid bodies.
They include a rotor in a Newtonian fluid and a fiber-reinforced material considered as a biphase material.
The formulation of the fluid-structure-interaction (FSI) in the first example by means of well-known finite element methods
leads to disadvantages in view of computational efficiency and stability if large rotations of embedded phases arise.
Well-known reasons are the difficult approximation of convective terms,
insufficient meshings of surface contacts as well as frequent remeshings of the phases.
The simulation of the solid-solid-interactions in the second example leads to long computing times,
because the mesh of the embedded phase determines the number of elements of the surrounding phase.
The reason is also the necessary sufficient approximation of surface contacts.
These disadvantages can be avoided by an immersed finite element method (IFEM).
Aims of this research project is the development and implementation of a new structure-preserving IFEM for dynamic, non-isothermal multiphase continua.
Here, fluids as well as solids are taken into account.
In order to consider also solids with rigid sections,
a noval non-isothermal rigid body formulation is developed,
which is directly based on a finite element method.
Thereby,
micropolar rotational degrees of freedom guarantee the rigidity of the finite element meshes pertaining to the rigid sections.
A special variational approach avoids the introduction of an Euler tensor,
and rigid sections of flexible solids can be defined by a simple declaration in the total mesh.
In this way,
thermomechanical couplings between non-isothermal rigid bodies, flexible solids and fluids can be simulated easily.
Well-known IFEM assume a fixed Eulerian mesh for the total continuum,
and consider Lagrangian meshes only for embedded phases.
This restriction will be abolished in the current project,
in order to simulate large deformations of non-isothermal multiphase solids by the IFEM.
But, also FSI simulations with an Eulerian mesh for a surrounding fluid will be improved by the structure-preserving IFEM.
There emerge new space-time approximations, which lead to an increasing numerical stability without user-defined stability parameters.
DFG GR 3297/7-1: Variational modelling and simulation of thermo-optochemo-dynamical coupling in liquid crystalline elastomers.
Modelling and simulation of coupled multiphysics problems in science and technology is currently an active field of research.
The goal is to model as exactly as possible unilateral and mutual actions between different fields of physics.
In this way, behaviour predictions and a targeted influencing of complex systems are possible.
An example is the induced deformation of a continuum by means of external multiphysical actions as a temperature change or ultraviolet light.
The deformation can causes the motion of the continuum itself, or the motion of bodies attached at the continuum boundary.
This is possible with liquid crystalline elastomers,
which can be largely deformed by a temperature field or ultraviolet light,
and are able to take up functions of more expensive and heavy motion mechanisms.
In the development of applications of such artifical materials during the design of devices, actuators and lightweight structures,
transient numerical simulations are more and more in use.
This reduce the number of time consuming and expensive experimental investigations,
and contribute to the conservation of natural resources and energy.
Especially for heterogeneous polymeric materials as liquid crystalline elastomers,
their targeted application can be developed and optimized by numerical simulations.
Here, it is preferable to apply a long-term stable simulation method,
which can be interfaced with existing finite element methods and therefore facilitates multibody simulations.
In this context,
variational material models and simulation methods supplemented by energy-momentum-consistent time integration algorithms are classified as long-term stable.
The aim of this research project is thus the variational modelling of the micro-macro-mechanical material behaviour of a liquid crystalline elastomer
by means of a noval generalized continuum based on functional formulations.
This enables a simulation with material specific mixed finite element methods,
and leads to a locking-free space discretization of the elastomeric parts.
In order to simulate multiphysically induced motions numerically stable and cpu-time efficient,
an energy-momentum-consistent time integration method is to be developed and implemented.
By using an automatic time step size control,
such algorithms supplimented by locking-free space discretizations perform less calculation steps,
and satisfy each balance law of a generalized continuum and coupled problem algorithmically exactly.
DFG GR 3297/6-1: Variational-based finite element simulation of fiber-reinforced materials with fiber bending stiffness in moving thermodynamical systems.
In scientific research and material science, computational simulations are more and more used for reducing costly and time consuming experimental investigations.
Especially in the case of composite materials, computational simulations provide the possibility to determine appropriate composites numerically first,
before these composite materials are produced physically.
In this way, many work hours and financial expense can be saved in the sample production,
because the number of samples will be reduced.
In order to determine appropriate composite materials by using the common finite element method,
there is a need for an exact as possible modelling of the appearing stress states in the considered parts.
In the case of the often appearing thin-walled composite structures,
the exact description of the bending behaviour in a computational simulation is therefore necessary.
Especially regarding dynamical loads, bending vibrations have to be predictable,
in order to know the right surrounding space and the appropriate support of the composite parts.
This requires to avoid locking effects in finite elements and the modelling of each material stiffness.
Thus, in the case of fiber-reinforced polymers with a predominant portion of solid fibers,
the computational modelling of a fiber bending stiffness is necessary.
Especially for parts subject to dynamical loads,
the modelling of an inertia influence by means of the fibers contributes to obtain meaningful computational results.
The goal of the submitted research project is the modelling of fiber bending or finite fiber diameter, respectively, in the scope of thermodynamics,
such that a dynamical simulation of the underlying material model is numerically exact, numerically stable and CPU-time efficient.
This is provided by the energy-momentum consistent time integration algorithms to be developed in this research project,
which are based on a locking-free space discretization and an automatic time-step size control.
DFG GR 3297/4-2: Physically consistent simulation of thermodynamics of fiber-reinforced plastics.
During the mechanical manufacturing of axisymmetric bodies with fiber-reinforced polymers (FRP) as pipes or shafts by using a winding process,
engineers use so-called rovings.
A roving denotes a bundle of parallel arranged continuous filaments.
The number of filaments lies in the range of multiple thousands.
Therefore, the diameter of a roving lies in the range of millimeters.
Moreover, rovings are used to produce woven composites.
A further manufacture technology is the Tailored-Fiber-Placement,
in which rovings are fixed by cords on a base material.
In these manufacture procedures, only rovings are used.
The anisotropic behaviour of the resulting metamaterials can be modelled as in the project GR 3297/4-1 by means of structural tensors.
But, an important feature of rovings is their bending stiffness.
The physical bending stiffness of rovings is not considered in the project GR 3297/4-1.
The fibers in this project are implicity assumed as infinitely thin.
The bending stiffness of the rovings leads to a large curvature under loading,
but to a small curvature at loading free boundaries.
Hence, a finite element mesh is simulated too flexible by means of the mixed finite element methods of project GR 3297/4-1.
Goal of the project GR 3297/4-2 is to take into account the correct physical bending stiffness of FRP parts made of rovings.
In this way, the energy-momentum schemes avoid artificial locking behaviour by using the mixed finite elements of project GR 3297/4-1,
but exactly simulate each physical stiffness of the FRP part.
Further, we aim at taking into account the material inhomogeneity,
because we obtain by rovings a multiscale problem with a macroscopic scale (the FRP part), a mesoscopic scale and microscopic scale (the roving).
DFG GR 3297/4-1: Physically consistent simulation of thermodynamics of fiber-reinforced plastics.
The use of fiber-reinforced plastics in lightweight constructions enables a cost reduced production of technical products,
and an energy efficient use of mobile systems.
Further, fiber-reinforced plastics lead to a very smooth running in mobile systems,
and therefore provide for a noise reduction in their environments.
Especially, heavy-weight metals are able to be replaced by fiber-reinforced plastics
due to an optimization of the fiber orientation with regard to tension strength of continuous components.
One aimes at an optimization in advance by using numerical simulations with reduced costs.
In order to obtain a possible real computational prediction of the motion and the material state of the considered component,
one have to use along with a thermodynamically consistent material model,
and adapted special continuum elements for a spatial discretization,
also a physically-consistent special time integration algorithm.
The goal of the submitted research project is the development of such physically consistent time integration algorithms
for finite motions and deformations of fiber-reinforced plastics,
which take into account the special modelling and spatial discretization of these anisotropic materials.
In order to guarantee the physical consistency during the simulation independent of the chosen spatial element and time step size,
the time integration algorithm has to be adapted to the material model and to the physically-consistent space discretization.
As special modelling of the elastic material behaviour,
it should be used the noval extit{polyconvex hyperelasticity} based on a polyconvex anisotropic free energy function,
in which the deformation gradient, its adjoint tensor and its determinant are considered as independent field variables.
The subsequent space-time discretization is based on the independency of these fields.
DFG GR 3297/2-(1,2): Structure-preserving time integrators for thermodynamics of nonlinear continua.
The steady increase of the performance of computers allows nowadays a computational analysis of thermodynamic processes
with a full consideration of mutual interactions of coupled physical fields.
Hence, these systems can be optimized much faster and cheaper.
In numerical simulations of thermodynamic systems, it is therefore important to apply time integrators,
which numerically exactly reproduce mutual interactions and, moreover, make the simulation more robust.
This is guaranteed if a time discretisation algorithmically reproduce the physical structure,
which means structural characteristics as balance laws and constitutive properties are fullfilled exactly in a discrete setting independent of the time discretisation.
In comparison to standard time integrators, structure preserving time integrators therefore are able to simulate processes also with a coarse time discretisation.
The aim of the submitted research project is to make the designed structure preserving time integrators more user-friendly.
The new algorithms should perform an individual adaption of the time discretisation in each coupled field subject to the structure preservation,
and at the same time a structure preserving adaption of the spatial finite element mesh.
The adaption of the time discretisation is obtained by determining the distribution of interpolation points of temporal shape functions in a given time step.
But this $-adaption will not be controlled by numerical errors as usual, but controlled by structure preservation.
The finite element adaption in space is based on a calculation of the nodal positions with the balance equation of material forces of the continuum (r-adaption).
