This work proposes a scheme for significantly reducing the computational complexity of discretized problems involving the non-smooth forward propagation of uncertainty by combining the adaptive hierarchical sparse grid stochastic collocation method (ALSGC) with a hierarchy of successively finer spatial discretizations (e.g. finite elements) of the underlying deterministic problem. To achieve this, we build strongly upon ideas from the Multilevel Monte Carlo method (MLMC), which represents a well-established technique for the reduction of computational complexity in problems affected by both deterministic and stochastic error contributions. The resulting approach is termed the Multilevel Adaptive Sparse Grid Collocation (MLASGC) method. Preliminary results for a low-dimensional, non-smooth parametric ODE problem are promising: the proposed MLASGC method exhibits an error/cost-relation of ε ∼ t −0.95 and therefore significantly outperforms the single-level ALSGC (ε ∼ t −0.65 ) and MLMC methods (ε t −0.5 ).
This work addresses the thermodynamically consistent formulation of bone remodeling as a fully implicit finite element material model. To this end, bone remodeling is described in the framework of thermodynamics for open systems resulting in a thermodynamically consistent constitutive law. In close analogy to elastoplastic material modeling, the constitutive equations are implicitly integrated in time and incorporated into a finite element weak form. A consistent linearization scheme is provided for the subsequent incremental non-linear boundary value problem, resulting in a computationally efficient description of bone remodeling. The presented model is suitable for implementation in any standard finite element framework with quadratic or higher-order element types. Two numerical examples in three dimensions are shown as proof of the efficiency of the proposed method.
<p>Numerical studies on integrity of the geological barriers in heat generating radioactive waste disposal remain a challenging topic involving modelling of thermal, hydraulic and mechanical (THM) processes within complex geometries, as well as particularly long simulation time intervals . Due to this, unfeasible computational complexity emerges for many three-dimensional problems, resulting in the need of further model assumptions and simplification for many types of simulation. To make use of results of such simulations reliably as a tool in the decision-making process, uncertainties introduced by the modelling have to be addressed in the framework of safety assessment.</p><p>Consequently, the system describing partial differential equations are dependent on a set of parameters, each parameter possibly subject to uncertainty resulting from reduced knowledge or imprecise measurement. The treatment of uncertainties introduces additional dimensions into the physical system, resulting in a dramatic increase of computational complexity for each parameter considered uncertain.</p><p>For general applicability, the method chosen for uncertainty quantification should be problem-independent, i.e. an arbitrary set of stochastic input data is propagated through the physical system, while the output is again a freely selectable quantity of interest. To this end, sampling-based methods like Monte-Carlo methods and stochastic collocation seem to be favourable.</p><p>Since a full stochastic model is never computable, it is amenable to include only the most sensitive parameters into stochastic analyses, retaining all other parameters as deterministic, in order to spend available computational power efficiently. With aim of finding such a suitable set of stochastic parameters, preliminary studies of simplified two-dimensional models with less complex geometries and a less complex TH-process seem to be appropriate.</p><p>In this contribution, a simplified two-dimensional model of a radioactive waste disposal in clayey rock is proposed, as a starting point, and its results of the thermal induced increase in pore water pressure is compared with more sophisticated and established models for a set of deterministic input parameters. It will be demonstrated that the simplified two-dimensional model is suitable for first stochastic investigation of pore water induced tensile or shear failure.</p><p>Subsequently, the results of different stochastic simulations for this model are presented, giving rise to a better understanding of stochastic modelling as well as stochastic post-processing in discretized problems for computational safety assessment of radioactive waste disposal. In detail, sensitivity of the quantity of interest to changes in the input parameters can be studied and in addition, worst-case scenarios within the parameter interval can be found. Given known probability density functions for each input parameter, probability of occurrence of each scenario as well as expected values and variances can be calculated.</p><p>&#160;</p><p>&#160;</p>
The problem of solving partial differential equations (PDEs) on manifolds can be considered to be one of the most general problem formulations encountered in computational multi-physics. The required covariant forms of balance laws as well as the corresponding covariant forms of the constitutive closing relations are naturally expressed using the bundle-valued exterior calculus of differential forms or related algebraic concepts. It can be argued that the appropriate solution method to such PDE problems is given by the finite element exterior calculus (FEEC). The aim of this essay is the exposition of a simple, efficiently-implementable framework for general hp-adaptivity applicable to the FEEC on higher-dimensional manifolds. A problem-independent spectral error-indicator is developed which estimates the error and the spectral decay of polynomial coefficients. The spectral decay rate is taken as an admissibility indicator on the polynomial order distribution. Finally, by elementary computational examples, it is attempted to demonstrate the power of the method as an engineering tool.
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