This paper introduces a new goal-oriented adap- tive technique based on a simple and effective post-process of the finite element approximations. The goal-oriented character of the estimate is achieved by analyzing both the direct problem and an auxiliary problem, denoted as adjoint or dual problem, which is related to the quantity of interest. Thus, the error estimation technique proposed in this paper would fall into the category of recovery-type explicit residual a posteriori error estimates. The procedure is valid for general linear quantities of interest and it is also extended to non-linear ones. The numerical examples demonstrate the efficiency of the proposed approach and discuss: 1) different error representations, 2) assessment of the dispersion error, and 3) different remeshing criteria.
Abstract
This paper introduces a new goal-oriented adap- tive technique based on a simple and effective post-process of the finite element approximations. The goal-oriented character of the estimate [...]
The Material Point Method (MPM) is widely used for challenging applica tions in engineering, and animation. The complexity of the method makes error estimation challenging. Error analysis of a simple MPM method is undertaken and the global error is shown to be first order in space and time for a widely-used variant of the method. Computational experiments illustrate the estimated accuracy.
Abstract
The Material Point Method (MPM) is widely used for challenging applica tions in engineering, and animation. The complexity of the method makes error estimation challenging. Error analysis of a simple MPM method is undertaken and the global error is shown to be first order in [...]
Numerical simulations and optimisation methods, such as mesh adaptation, rely on the accurate and inexpensive use of error estimation methods. Adjoint-based error estimation is the most accurate method, and generally the most costly. A strong contributor to this cost is the need to compute a higher resolution adjoint solution, using time dependent information. Here, recontruction methods applied to the primal and adjoint solutions are proposed to alleviate both the storage footprint of the primal problem and the adjoint computational cost. The method is compared to reference error estimators on an unsteady Burgers’ equation using the method of manufactured solutions. Two reconstruction methods, a proper orthogonal decomposition and a static convolutional neural network were used to demonstrate both the computational cost reduction and the potential for the reduction of the storage footprint of the primal problem. When reconstruction methods are applied to the primal problem, one can use both proposed approaches to reduce the footprint of the solution and reconstruct the effectively compressed primal solution to be recalled for the adjoint solution. The second approach consists in solving an adjoint solution from a coarse primal solution and using reconstruction methods to obtain a higher resolution adjoint solution, necessary for output error estimation and mesh adaptation. The obtained results give great confidence in the use of reconstruction methods for the reduction of both computational cost and storage requirements of adjoint-based error estimation, and goal-oriented mesh adaptation.
Abstract
Numerical simulations and optimisation methods, such as mesh adaptation, rely on the accurate and inexpensive use of error estimation methods. Adjoint-based error estimation is the most accurate method, and generally the most costly. A strong contributor to this cost is the need [...]