A Priori error analyses of a stabilized discontinuous Galerkin method

AbstractWe introduce a new stabilized discontinuous Galerkin method within a new function space setting, that is closely related to the discontinuous Galerkin formulation by Oden, Babuška and Baumann [1], but involves an extra stabilization term on the jumps of the normal fluxes across the element interfaces. The formulation satisfies a local conservation property and we prove well posedness of the new formulation. A priori error estimates are derived, which are verified by 1D and 2D experiments on a reaction-diffusion type model problem

Dynamic splinting home therapy for toe walking: a case report

Serial casting is frequently prescribed for toe-walking but that does not allow continued physical therapy (PT). This report described a child and family who chose dynamic splinting (DS) with concurrent PT for treatment. The patient presented with right hemiparesis; below average motor skills and a gait pattern of toe contact (without ankle foot orthosis). Four months of PT plus 6 hours/night of DS as home therapy, the patient's passive dorsiflexion increased 14° and she gained the ability to walk in "flat foot" contact without the Ankle Foot Orthosis. This concurrent treatment achieved improved gait pattern and strength training not possible with casting

Methods of Approximation in hpk Framework for ODEs in Time Resulting from Decoupling of Space and Time in IVPs

This is the published version. Copyright © 2011 Scientific Research PublishingThe present study considers mathematical classification of the time differential operators and then applies methods of approximation in time such as Galerkin method (GM ), Galerkin method with weak form (/GMWF ), Petrov-Galerkin method (PGM), weighted residual method (WRY ), and least squares method or process (LSM or LSP ) to construct finite element approximations in time. A correspondence is estab- lished between these integral forms and the elements of the calculus of variations: 1) to determine which methods of approximation yield unconditionally stable (variationally consistent integral forms, VC ) com- putational processes for which types of operators and, 2) to establish which integral forms do not yield un- conditionally stable computations (variationally inconsistent integral forms, VIC). It is shown that varia- tionally consistent time integral forms in hpk framework yield computational processes for ODEs in time that are unconditionally stable, provide a mechanism of higher order global differentiability approxima- tions as well as higher degree local approximations in time, provide control over approximation error when used as a time marching process and can indeed yield time accurate solutions of the evolution. Numerical studies are presented using standard model problems from the literature and the results are compared with Wilson’s method as well as Newmark method to demonstrate highly meritorious features of the pro- posed methodology

Goal-oriented adaptivity using unconventional error representations for the multi-dimensional Helmholtz equation

In goal‐oriented adaptivity, the error in the quantity of interest is represented using the error functions of the direct and adjoint problems. This error representation is subsequently bounded above by element‐wise error indicators that are used to drive optimal refinements. In this work, we propose to replace, in the error representation, the adjoint problem by an alternative operator. The main advantage of the proposed approach is that, when judiciously selecting such alternative operator, the corresponding upper bound of the error representation becomes sharper, leading to a more efficient goal‐oriented adaptivity. While the method can be applied to a variety of problems, we focus here on two‐ and three‐dimensional (2‐D and 3‐D) Helmholtz problems. We show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones and lead to a more robust p‐adaptive process. We also provide guidelines for finding operators delivering sharp error representation upper bounds. We further extend the results to a convection‐dominated diffusion problem as well as to problems with discontinuous material coefficients. Finally, we consider a sonic logging‐while‐drilling problem to illustrate the applicability of the proposed method.V. Darrigrand, A. Rodriguez-Rozas and D. Pardo were partially funded by the Projects of the Spanish Ministry of Economy and Competitiveness with reference MTM2013-40824-P, MTM2016-76329-R (AEI/FEDER, EU), MTM2016-81697-ERC and the Basque Government Consolidated Research Group Grant IT649- 13 on “Mathematical Modeling, Simulation, and Industrial Applications (M2SI)”. A. Rodriguez-Rozas and D.Pardo were also partially funded by the BCAM “Severo Ochoa” accreditation of excellence SEV-2013-0323 and the Basque Government through the BERC2014-2017 program. A. Rodriguez-Rozas acknowledges support from Spanish Ministry under Grant No. FPDI- 2013-17098. I. Muga was partially funded by the FONDECYT project 1160774. The first four authors were also partially funded by the European Union’s Horizon 2020, research and innovation program under the Marie Sklodowska-Curie grant agreement No 644202. Serge Prudhomme is grateful for the support by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada

Normative EMG activation patterns of school-age children during gait

Gait analysis is widely used in clinics to study walking abnormalities for surgery planning, definition of rehabilitation protocols, and objective evaluation of clinical outcomes. Surface electromyography allows the study of muscle activity non-invasively and the evaluation of the timing of muscle activation during movement. The aim of this study was to present a normative dataset of muscle activation patterns obtained from a large number of strides in a population of 100 healthy children aged 6-11 years. The activity of Tibialis Anterior, Lateral head of Gastrocnemius, Vastus Medialis, Rectus Femoris and Lateral Hamstrings on both lower limbs was analyzed during a 2.5-min walk at free speed. More than 120 consecutive strides were analyzed for each child, resulting in approximately 28,000 strides. Onset and offset instants were reported for each observed muscle. The analysis of a high number of strides for each participant allowed us to obtain the most recurrent patterns of activation during gait, demonstrating that a subject uses a specific muscle with different activation modalities even in the same walk. The knowledge of the various activation patterns and of their statistics will be of help in clinical gait analysis and will serve as reference in the design of future gait studie