In the thesis the advanced way of numerical modeling of dynamic contact problems in the time space was presented. First the formulation of the space-time finite element method was presented. Both the displacement and velocity formulation was developed. The space-time approach presented in the thesis can be considered as the extension of the finite element method, which is usually applied to space. In FEM time variable requires the separate treatment. The discretization is uncoupled. In the space-time finite element method the discretization is applied at the same time to the space and to time, ie. to the time space. The spatial and time discretization are coupled. It was shown that in the particular case the presented approach is identical with the finite element method applied to spatial derivatives, and the difference method applied to time derivatives. In the general case the structure can be split into finite subdomains in the non stationary way. Properties of the method were examined in details. The amplitude error, phase error and numerical damping were tested and compared with other methods. Special shapes of space-time finite elements (simplex-shaped) decouple the resulting system of algebraic equations. It eliminates the triangulation stage in the solution process. Physical properties of the simplex shaped mesh are interesting since the information flow in the direction of slope edges in the space-time mesh is limited. It enables to treat infinite uni-dimensional structures subjected to a fastly mooving load by reducing the infinite system of finite elements to a set of few elements only.
The soft way method was used for contact conditions. The contact conditions with singular velocity in the contact were modified to obtain continuous displacement and velocity functions. The displacement is computed in a consistent way from the velocity. Both the consistency and continuity were achieved by the reduction of the velocity one time step before the penetration. Another way proposed for contact modelling is directly derived from the space--time finite element modeling. The time step is divided in the selected part of the somain, in our case in the contact region. Triangular space-time elements were used to this purpose. Presented technique was applied in the simulation of the corrugation generation in the train wheel. The solution of the problem, intensively investigated by researchers, showed the wave nature of the wheel polygonization.
The adaptivity of the mesh is the fundamental advantage of the space--time approach. The r-adaptive technique enables to relocate the mesh joints, with the limited speed, however. The h-adaptive method allows to add or remove nodes in space and in this way to refine or coarsen the spatial mesh. Mesh adaptivity applied to structural dynamics should fulfil strong restrictions. Since each perturbation in the mesh or in nodal values considerably changes the response of the modified system, the simple interpolation for incorporated joints fails. In the thesis the space--time triangular elements described by velocities enables successful h--adaptation for vibrating structures, not only in low frequency range but also in wave problems. --- (full text: *.ps.gz)
Thermal waves in heat conductor and their numerical
modelling.
Numerical solutions by means of
the space--time finite element method to initial-boundary
value problems for a hyperbolic model of heat conduction, are
obtained. The heat conduction description is based on a concept
a rigid conductor with a scalar internal state variable, that leads
to a modified Fourier law. The obtained results are compared with
existing experimental data
know for semi-conductor crystals at low temperature.
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Space-time finite element
formulation for the dynamical evolutionary process
In the paper a new formula for the space time finite
element approach to dynamic solution of solid is developed.
One degree of freedom system was
analyzed to find an unconditionally stable scheme of
integration. Relations are expressed in terms of
velocity. The geometry of the domain analysed is updated
in every time step. The procedure can be efficient for
geometrically non linear problems of mechanics of continuum.
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On existence and numerical approach
of the solution for rigid-viscoplastic materials with friction
This paper deals with the study of a boundary value problem for
rigid-viscoplastic materials submitted to large deformations.
The boundary conditions considered are unilateral conditions with
friction. A velocity formulation is proposed and an existence and
uniqueness result is obtained. Two numerical methods of approximation
based on the finite element method are also given and the interest
for the space-time finite element method is underlined. Two concrete
numerical examples concerning the impact of a cylinder with a rigid
surface and the plane strain upsetting of a cylindrical tube are also
presented.
New formulation of the space-time
finite element method for problems of evolution
In the paper the space-time finite element method was developed
for problems of evolution. The equilibrium equations were determined
in terms of velocity. Non-stationary partition into spatial
finite elements, which arises from the evolution of the shape of
material, was assumed. Properties of the solution scheme,
particularly convergence and stability, depends on
the form of the distribution of virtual velocity.
The system of one degree of freedom, described both by
linear and non linear differential equation, was investigated.
The damping of higher-mode oscillations and the amplitude and
phase error was estimated. The solution of testing and
real problems was performed. High efficiency of the proposed
method for complex problems, also with internal contact,
was proved.--- (full text: *.ps.gz)
On the space-time element method applied
to the dynamics of vehicle - road interaction
The aim of the paper is to verify one of the transportation
systems previously introduced by K. Popp and R. Bogacz and partly
investigated analytically by the above authors as
well as T. Krzyzynski, Ostermayer, Riemer, Sperler and others.
Since the known analytical solution is limited to the steady state motion,
there is a need to investigate the vehicle behaviour during start up,
breaking the process and during passing the region of instability.
The paper is devoted to the application of the Space-Time Element
Method to the simulation of the vehicle - road interaction
modelled as a hybrid system with two points of interaction.
The numerical investigation will also be presented.
Dynamic contact problem by means of the space-time
element method
The space-time element approach was applied in the
investigation of the rolling contact problem. The a-posteriori
error estimation enabled the mesh condensation in regions of high
stress gradient and coarsening aside. Refined parts move together
with the roller ensuring the low error level. Nonlinear
incremental formulas for the space-time solution scheme was
derived. Tetrahedral space-time elements for plane strain allow to
obtain triangular form of the coefficient matrix directly during
the global matrix assembly. The joint-by-joint procedure is a
natural way of the solution then.
Adaptive space-time elements in the dynamic elastic-viscoplastic problem
An adaptive technique for the solution of the dynamic elastic-viscoplastic
problem has been developed. The mesh modification is performed by the use
of the space-time element method according to the error estimation.
The number of joints is preserved and the mesh is refined in regions
of high stress gradients. It enables the size of the problem to be reduced
and increases the speed of computations. The incremental procedure in the case
of small time step allows the nonlinear path iteration to be associated
with the time marching scheme. The remesh and remap problems related
to stresses are described. Numerical examples of a plane strain
rolling contact problem and collision of the plane strain object prove
the efficiency of the approach.
Mesh r-adaptation in structural dynamics
The adaptive technique for the solution of the dynamic elastic-viscoplastic
problem has been developed. The mesh modification is performed by the use
of the space-time element method according to the error estimation.
The approximation error is estimated in each time
step and depending on its value the mesh is modified.
So called r-adaptation preserves the number of joints
and moves the nodal points toward the domains of higher
stress gradients.
It enables to reduce the size of the problem and
increases the speed of computations. The incremental procedure in the case
of small time step allows to associate the nonlinear path iteration
with the time marching scheme. The remesh and remap problems related
to stresses have been described.
In the presented approach the mesh adaptation is performed by the space-time element method (STEM) and is applied to structural dynamics. The simplex-shaped elements are applied. It allows to obtain directly the system of separated algebraic equations that can be solved with the joint-by-joint scheme. It means that joints of the spatial mesh taken in two successive whiles are connected and in this way the space-time subdomains can be determined. Since the joints have different coordinates in whiles that limit the time layer at the top and the bottom, the space-time elements have non-rectangular forms.
Numerical examples of plane strain rolling contact problem and plane strain collision prove the efficiency of the approach.
Adaptive mesh in dynamic problem by the space-time approach
Stress fields varying in time are typical for dynamic wave
problems. Nonclassic problems involve changing of structure
properties, especially wave reflection zones or dissipative zones.
Stress field propagation requires a variable mesh that allows one to
approach the phenomenon with the smallest error in each time step.
The space-time approximation of the differential equation of
motion enables the modification of the spatial partition into
finite elements in a continuous way. Error estimation was the
reason to refine and coarsen the spatial partition, moving the
nodes towards the zone of higher error. Applying the simplex-shaped
space-time elements one can gain the triangular form of
coefficient matrix directly in the element matrix assembly
process. Consistent characteristic matrices are used. The approach
presented was successfully applied for bar, beam and plane strain
analysis. The method is more powerful for materially nonlinear
cases for which element matrices should be calculated in each time
step. Good accuracy of the movable mesh approach was proved in
several testing examples.
Space-time element method in structural dynamics
- state-of-the art
The paper deals with the recent developments of the
space-time element method for vibration analysis.
Discrete methods applied to date in structural dynamics
use the spatial discretization independently of the
time integration procedure. It limits applications of
such an approach. The full space-time approximation can
be considered as an extension of the finite element
method over the time domain and it allows to treat
spatial variables in the same way as the time variable.
Nonstationary discretization, adaptive techniques,
directly obtained joint-by-joint procedure are not the
only positive features of the space-time finite element
approach. Although the additional time variable in the
shape functions is considered and the resulting element
matrices are greater than static stiffness and mass
matrices, the cost of the solution algorithm is
comparative with other numerical methods. Some testing
examples prove the efficiency of the method.
Kinematic approach for dynamic contact
problems - the geometrical soft way method
A way of taking into account the
geometrical constraints in evolution problems of solid systems
that limit the possibility of motion by the history of variation
of the velocity field is developed in the paper.
A formulation that can be adapted particularly to a numerous
problems of solid systems subjected to dynamics effects with
large deformations, large displacements, large rotations
is described.
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The soft way method and the velocity formulation
In the paper the new approach to dynamic contact problems is described.
The velocity formulation was assumed and a new time integration scheme
was elaborated. The space--time finite element method used in derivation
enables the control of the accuracy (order of the error) and stability.
Methods for the solution of contact problems were discussed. Discretized
approach, prepared for large displacements and large rotations enabled
to solve real engineering problems in a relatively short time.
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