Analysis of Dual Null Field Methods for Dirichlet Problems of Laplace’s Equation in Elliptic Domains with Elliptic Holes: Bypassing Degenerate Scales

Dual techniques have been used in many engineering papers to deal with singularity and ill-conditioning of the boundary element method (BEM). Our efforts are paid to explore theoretical analysis, including error and stability analysis, to fill up the gap between theory and computation. Our group provides the analysis for Laplace’s equation in circular domains with circular holes and in this paper for elliptic domains with elliptic holes. The explicit algebraic equations of the first kind and second kinds of the null field method (NFM) and the interior field method (IFM) have been studied extensively. Traditionally, the first and the second kinds of the NFM are used for the Dirichlet and Neumann problems, respectively. To bypass the degenerate scales of Dirichlet problems, the second and the first kinds of the NFM are used for the exterior and the interior boundaries, simultaneously, called the dual null field method (DNFM) in this paper. Optimal convergence rates and good stability for the DNFM can be achieved from our analysis. This paper is the first part of the study and mostly concerns theoretical aspects; the second part is expected to be devoted to numerical experiments.

Increase in weighting of vision vs. proprioception associated with force field adaptation

Hand position can be encoded by vision, via an image on the retina, and proprioception (position sense), via sensors in the joints and muscles. The brain is thought to weight and combine available sensory estimates to form an integrated multisensory estimate of hand position with which to guide movement. Force field adaptation, a form of cerebellum-dependent motor learning in which reaches are systematically adjusted to compensate for a somatosensory perturbation, is associated with both motor and proprioceptive changes. The cerebellum has connections with parietal regions thought to be involved in multisensory integration; however, it is unknown if force adaptation is associated with changes in multisensory perception. One possibility is that force adaptation affects all relevant sensory modalities similarly, such that the brain’s weighting of vision vs. proprioception is maintained. Alternatively, the somatosensory perturbation might be interpreted as proprioceptive unreliability, resulting in vision being up-weighted relative to proprioception. We assessed visuo-proprioceptive weighting with a perceptual estimation task before and after subjects performed straight-ahead reaches grasping a robotic manipulandum. Each subject performed one session with a clockwise or counter-clockwise velocity-dependent force field, and one session in a null field to control for perceptual changes not specific to force adaptation. Subjects increased their weight of vision vs. proprioception in the force field session relative to the null field session, regardless of force field direction, in the straight-ahead dimension (F1,44 = 5.13, p = 0.029). This suggests that force field adaptation is associated with an increase in the brain’s weighting of vision vs. proprioception.

Green’s Function Problem of Laplace Equation with Spherical and Prolate Spheroidal Boundaries by Using the Null-Field Boundary Integral Equation

A systematic approach of using the null-field integral equation in conjunction with the degenerate kernel and eigenfunction expansion is employed to solve three-dimensional (3D) Green’s functions of Laplace equation. The purpose of using degenerate kernels for interior and exterior expansions is to avoid calculating the principal values. The adaptive observer system is addressed to employ the property of degenerate kernels in the spherical coordinates and in the prolate spheroidal coordinates. After introducing the collocation points on each boundary and matching boundary conditions, a linear algebraic system is obtained without boundary discretization. Unknown coefficients can be easily determined. Finally, several examples are given to demonstrate the validity of the present approach.