1,721,194 research outputs found
The Weak Coupling Method for Coupling Continuum Mechanics with Molecular Dynamics
Full text linkFor the global behavior of solids in structural mechanics of nonlinear processes, local effects on the atomistic level play an important role. Often a direct numerical simulation of the macroscopic behavior by a complete resolution of the microscale is for computational reason not possible. Thus, employing a multiscale strategy for an efficient and accurate modelling seems favorable since by separating the problem into two different frameworks, the accuracy of a fine scale model can be combined with the advantages of a computationally efficient model. More precisely a comparably small region of atoms e.g. surrounding the tip of a crack is modelled by molecular dynamics. Outside of this region, we take advantage of the fact that the displacement is almost homogeneous and can thus be modelled efficiently by a linear elastic continuum dynamical simulation. Clearly, both scales offer fundamentally different descriptions of the matter and they use different simulation methods. Whereas on the continuum scale the finite element method and a function space setting is used, the molecular dynamics is based on the movement of particles in the Euclidean space. Additionally, dynamical simulations with a transition zone (handshake region) between atomistic systems and the coarser finite element mesh suffer from unwanted (spurious) reflections, since the finite element method can not represent short wave length vibrational modes. Here a completely new approach is presented, which takes advantage of an infinite dimensional function space for the information transfer between the scales. Starting from a handshake region, the key idea is to construct a transfer operator between the different scales. This transfer operator is based on local averaging taken values. In order to construct the local weight functions, a partition of unity is assigned to the molecular degree of freedom. This allows us to decompose the micro scale displacement in the handshake region into a small and large wave number part by means of a weighted projection. In the first instance, this function space oriented interpretation of the atomistic displacement is applied in the context of a completely overlapping decomposition. More precisely, we consider the case, when the domain of the handshake region is conform with the domain of the molecular dynamics. In order to identify the displacements pertaining to the atomistic or continuum level respectively, we employ a multiscale decomposition. In particular, we decompose the micro scale displacement into a "low frequency'' and a "high frequency'' part in a weak sense. This new approach is also used in the context of a partly overlapping decomposition. Therein, the coarse and the fine scale simulation are matched by constraining the two displacements in the handshake region. The key issue in this context is, that our function space oriented approach allows us to interpret the constraints in a weak sense. Thus the "low frequent'' part can be captured by the coarse scale, whereas the "high frequent'' part of the displacement which has no meaning on the coarse scale is damped in the handshake region. Moreover numerical examples in 1d,2d and 3d show that this approach allows molecular displacements for entering into the continuum domain and the other way round flawlessly
Quadrature methods for elliptic PDEs with random diffusion
Full text linkIn this thesis, we consider elliptic boundary value problems with
random diffusion coefficients. Such equations arise in many
engineering applications, for example, in the modelling of
subsurface flows in porous media, such as rocks.
To describe the subsurface flow, it is convenient to use
Darcy's law. The key ingredient in this approach is the hydraulic
conductivity. In most cases, this hydraulic conductivity is approximated
from a discrete number of measurements and, hence, it is common to
endow it with uncertainty, i.e. model it as a random field.
This random field is usually characterized
by its mean field and its covariance function.
Naturally, this randomness propagates through the model which
yields that the solution is a random field as well.
The thesis on hand is concerned with the effective computation
of statistical quantities of this random solution, like the expectation,
the variance, and higher order moments.
In order to compute these quantities, a suitable representation of the
random field which describes the hydraulic conductivity needs to be
computed from the mean field and the covariance function.
This is realized by the Karhunen-Loeve expansion which
separates the spatial variable and the stochastic variable. In general, the
number of random variables and spatial functions used in this expansion
is infinite and needs to be truncated appropriately.
The number of random variables which are required depends on the
smoothness of the covariance function and grows with the desired accuracy.
Since the solution also depends on these random variables, each moment
of the solution appears as a high-dimensional Bochner integral over the
image space of the collection of random variables. This integral has to be
approximated by quadrature methods where each function evaluation
corresponds to a PDE solve.
In this thesis, the Monte Carlo, quasi-Monte Carlo, Gaussian tensor product, and
Gaussian sparse grid quadrature is analyzed to deal with this high-dimensional
integration problem.
In the first part, the necessary regularity requirements of the integrand and
its powers are provided in order to guarantee convergence of the different
methods.
It turns out that all the powers of the solution depend, like the solution itself,
anisotropic on the different random variables which means in this case that
there is a decaying dependence on the different random variables.
This dependence can be used to overcome, at least up to a certain extent, the
curse of dimensionality of the quadrature problem.
This is reflected in the proofs of the convergence rates of the different
quadrature methods which can be found in the second part of this thesis.
The last part is concerned with multilevel quadrature approaches to keep
the computational cost low. As mentioned earlier, we need to solve a partial
differential equation for each quadrature point.
The common approach is to apply a finite element approximation scheme on
a refinement level which corresponds to the desired accuracy.
Hence, the total computational cost is given by the product of the number
of quadrature points times the cost to compute one finite element solution
on a relatively high refinement level.
The multilevel idea is to use a telescoping sum decomposition of the quantity
of interest with respect to different spatial refinement levels and use
quadrature methods with different accuracies for each summand.
Roughly speaking, the multilevel approach spends a lot of quadrature points
on a low spatial refinement and only a few on the higher refinement levels.
This reduces the computational complexity but requires further regularity
on the integrand which is proven for the considered problems in this thesis
Trial methods for Bernoulli's free boundary problem
Full text linkFree boundary problems deal with solving partial differential equations in a domain,
a part of whose boundary is unknown – the so-called free boundary. Beside
the standard boundary conditions that are needed in order to solve the partial
differential equation, an additional boundary condition is imposed at the free
boundary. One aims thus to determine both, the free boundary and the solution
of the partial differential equation.
This thesis is dedicated to the solution of the generalized exterior Bernoulli
free boundary problem which is an important model problem for developing algorithms
in a broad band of applications such as optimal design, fluid dynamics,
electromagnentic shaping etc. Due to its various advantages in the analysis and
implementation, the trial method, which is a fixed-point type iteration method,
has been chosen as numerical method.
The iterative scheme starts with an initial guess of the free boundary. Given
one boundary condition at the free boundary, the boundary element method is
applied to compute an approximation of the violated boundary data. The free
boundary is then updated such that the violated boundary condition is satisfied
at the new boundary. Taylor’s expansion of the violated boundary data around
the actual boundary yields the underlying equation, which is formulated as an
optimization problem for the sought update function. When a target tolerance is
achieved the iterative procedure stops and the approximate solution of the free
boundary problem is detected.
How efficient or quick the trial method is converging depends significantly
on the update rule for the free boundary, and thus on the violated boundary
condition. Firstly, the trial method with violated Dirichlet data is examined and
updates based on the first and the second order Taylor expansion are performed.
A thorough analysis of the convergence of the trial method in combination with
results from shape sensitivity analysis motivates the development of higher order
convergent versions of the trial method. Finally, the gained experience is
exploited to draw very important conclusions about the trial method with violated
Neumann data, which is until now poorly explored and has never been
numerically implemented
A Unifying Theory for Nonlinear Additively and Multiplicatively Preconditioned Globalization Strategies : Convergence Results and Examples From the Field of Nonlinear Elastostatics and Elastodynamics
Full text linkNonlinear right preconditioned globalization strategies for the solution of nonlinear programming problems of the following kind where is a convex set of admissible solutions, , and , sufficiently smooth, are presented. Preconditioned globalization strategies are traditional Linesearch or Trust-Region strategies in combination with a nonlinear update operator which results from a nonlinear solution process for smaller, but related, nonlinear programming problems. We will formulate conditions on this abstract operator, in order to ensure global convergence, i.e., convergence to first-order critical points, of the resulting method. In addition, we introduce particular implementations of this abstract operator, i.e., nonlinear multiplicatively preconditioned Trust-Region (MPTS) and Linesearch strategies (MPLS), as well as nonlinear additively preconditioned Trust-Region (APTS) and Linesearch (APLS) strategies. As it turns out, these additive strategies are novel parallel, locally adaptive and robust solution methods for nonlinear programming problems. Moreover, the MPTS strategy generalizes the RMTR concepts in [GK08] in order to allow also for the application of alternating nonlinear domain decomposition methods. On the other hand, the MPLS method simplifies and generalizes the concepts in [WG08] giving rise to a novel solution strategy for pointwise constrained nonlinear programming problems. The respective nonlinear solution strategies are analyzed and global convergence is shown. In addition, global convergence is also shown for combined nonlinear additively and multiplicatively preconditioned Trust-Region and Linesearch strategies. Moreover, we show the efficiency and reliability of these methods in the context of problems arising from the field of nonlinear elasticity in 3d. Particular emphasis has been placed on the formulation and analysis of the resulting minimization problems. Here, we show that these problems satisfy the assumptions stated to show convergence of the respective preconditioned globalization strategies. Moreover, various elasto-static and elasto-dynamic examples are presented in order to compare the convergence rates and runtimes of the different strategies
Numerical methods for boundary value problems on random domains
Full text linkIn this thesis, we consider the numerical solution of
elliptic boundary value problems on random domains.
The underlying domain is modelled
via a random vector field which is given by its mean
and its covariance.
Having these statistics of the random perturbation at
hand, we aim at determining the related statistics of
the random solution.
To that end, we propose the domain mapping method
on the one hand and the perturbation method on the
other hand.
For the domain mapping method, we have to compute the
random vector field's Karhunen-Loève expansion.
For this purpose, we compare cluster methods, namely
the adaptive cross approximation and the fast multipole
method, and the pivoted Cholesky decomposition.
After this, we show regularity results for the random
solution dependent on the decay of the random vector
field's Karhunen-Loève expansion. These results are used
to employ a Quasi-Monte Carlo quadrature for the
approximation of mean and variance.
For the perturbation method, we linearize the random
solution's dependence on the vector field by means of
a shape Taylor expansion. This approach yields a single
partial differential equation for the approximation of
the mean and a tensor product partial differential
equation for the approximation of the covariance. The latter
is solved efficiently with the aid of the sparse tensor
product combination technique
An open-loop approach to study the stochastic properties
Full text linkRandomness is an inevitable aspect of biological networks. It has been long accepted that variability of components in a network can propagate throughout the network. In this thesis, we introduce a method that allows us to decompose the total variability of a single component into individual contributions from the other components in a network. Our method of noise decomposition helps us investigate key parameters and their relative impact on the total normalized noise and also allows us to illustrate the importance of different
system modifications by adding or omitting biological processes. With our generally applicable noise decomposition method, we are able to determine the strength of individual correlations induced by different co-regulation processes that connect different components
of a network. In bistable systems, variability can occur through stochastic transitions from one steady state to another. Noise induced transitions between two steady states are difficult to calculate due to the intricate interplay between nonlinear dynamics and noise in bistable positive feedback loops. We open multicomponent feedback loops at the slowest variables in order to calculate the transition rates from one steady state to another. By reclosing the feedback loop, we calculate the mean first passage time (MFPT) using the Fokker-Planck equation. It is important to emphasize that the accurate
approximation of the open-loop results is not a sufficient condition for a good prediction of the MFPT. We show that only the opening at the slowest variable warrants an accurate prediction of MFPT. Multiplicative interactions among different components can introduce correlations among noises. We show that the introduced correlations affect the mean and variance of the open loop function and consequently increase the transition rate between two steady states in the closed-loop system. Our results indicate that the open-loop approach can contribute to the theoretical prediction of the MFPT. The theoretical results are shown to be in good agreement with the results of stochastic simulation
Hierarchical matrix techniques for partial differential equations with random input data
Full text linkWe consider the solution of elliptic partial differential equations with random input data. In particular, we are interested in the mean and the correlation of the solution of these partial differential equations. Once the correlation is available, the variance can be computed from its diagonal.
If the dependence of the solution on the random input data is linear and the mean of the input data is available, the mean of the solution can be computed whenever the corresponding partial differential equation can be solved. When the correlation of the input data is available, the correlation of the solution is given as the solution of a higher dimensional partial differential equation in the product domain. A Galerkin discretization yields a matrix equation with a typically densely populated right-hand side. This and the higher dimension of the problem makes the equation prohibitively expensive to solve.
Since existing approaches to these correlation equations are known to struggle for roughly correlated input data, we use hierarchical matrices and their corresponding arithmetic to represent and solve the matrix equation. The regularity assumptions required for hierarchical matrices can be justified on sufficiently smooth domains. The feasibility of the approach is illustrated for finite element and boundary element discretizations and several specialities of hierarchical matrices for finite element discretizations are discussed. An almost linear scaling with respect to the dimension of the used finite element spaces is verified.
The nonlinear dependence of the solution on the random input data is exemplarily discussed for the case of random domains. Extending previous perturbation approaches, we can linearize the problem and can compute the mean and the correlation up to third order accuracy in almost linear time. We discuss that these spaces can become even fourth order accurate under certain circumstances. A full convergence analysis is presented, which shows that higher order boundary elements are required for the solution of the problem. Therefore, we develop a fast multipole method for higher order boundary element methods on parametric surfaces
A stochastic algorithm for the identification of solution spaces in high-dimensional design spaces
Full text linkThe volume of an axis-parallel hyperbox in a high-dimensional design space is to be maximized under the constraint that the objective values of all enclosed designs are below a given threshold. The hyperbox corresponds to a Cartesian product of intervals for each input parameter. These intervals are used to assess robustness or to identify relevant parameters for the improvement of an insufficient design.
A related algorithm which is applicable to any non-linear, high-dimensional and noisy problem with uncertain input parameters is presented and analyzed. Analytical solutions for high-dimensional benchmark problems are derived. The numerical solutions of the algorithm are compared with the analytical solutions to investigate the efficiency of the algorithm. The convergence behavior of the algorithm is studied. The speed of convergence decreases when the number of dimensions increases. An analytical model describing this phenomenon is derived. Relevant mechanisms are identified that explain how the number of dimensions affects the performance. The optimal number of sample points per iteration is determined depending on the preference for fast convergence or a large volume. The applicability of the method to a high-dimensional and non-linear engineering problem from vehicle crash analysis is demonstrated. Moreover, we consider a problem from a forming process and a problem from the rear passenger safety.
Finally, the method is extended to minimize the effort to turn a bad into a good design. We maximize the size of the hyperbox under the additional constraint that all parameter values of the bad design are within the resulting hyperbox except for a few parameter values. These parameters are called key parameters because they have to be changed to lie within their desired intervals in order to turn the bad into a good design. The size of the intervals represents the tolerance to variability caused, for example, by uncertainty. Two-dimensional examples are presented to demonstrate the applicability of the extended algorithm. Then, for a high-dimensional, non-linear and noisy vehicle crash design problem, the key parameters are identified. From this, a practical engineering solution is derived which would have been difficult to find by alternative methods
An adaptive wavelet method for the solution of boundary integral equations in three dimensions
Full text linkIn science and engineering one often comes across partial differential equations in three dimensions, some of which can be formulated as boundary integral equations on the boundary of the three-dimensional domain of interest. With this approach the dimensionality of the problem can be reduced by one dimension and the interior as well as the exterior problem can be solved. However, this advantage does not come entirely without cost, as the involved matrices are dense. By using a wavelet scheme many matrix entries become sufficiently small such that they can be neglected without compromising the convergence rate of the underlying Galerkin scheme. In this thesis we go a step further and use an adaptive wavelet approach, meaning that specific parts of the geometry will be resolved with much detail, while other parts can stay coarse. After we have introduced the necessary theoretical foundation on boundary integral equations, wavelets and adaptive wavelet schemes, we present the details on the implementation followed by several numerical results. In the final chapter of this thesis we present the concept of goal-oriented error estimation, again followed by numerical results
- …
