We present the frequency-domain boundary element formulation for solving surface second-harmonic generation from nanoparticles of virtually arbitrary shape and material. We use the Rao-Wilton-Glisson basis functions and Galerkin's testing, which leads to very accurate solutions for both near and far fields. This is verified by a comparison to a solution obtained via multipole expansion for the case of a spherical particle. The frequency-domain formulation allows the use of experimentally measured linear and nonlinear material parameters or the use of parameters obtained using ab-initio principles. As an example, the method is applied to a non-centrosymmetric L-shaped gold nanoparticle to illustrate the formation of surface nonlinear polarization and the second-harmonic radiation properties of the particle. This method provides a theoretically well-founded approach for modelling nonlinear optical phenomena in nanoparticles.

Abstract. A homology and cohomology solver for finite element meshes is represented. It is an integrated part of the finite element mesh generator Gmsh. We demonstrate the exploitation of the cohomology computation results in a finite element solver, and use an induction heating problem as a working example. The homology and cohomology solver makes the use of a vectorscalar potential formulation straightforward. This gives better overall performance than a vector potential formulation. Cohomology computation also clarifies the lumped parameter coupling of the problem and enables the user to obtain useful post-processing data as a part of the finite element solution.Key words. homology computation, cohomology computation, finite element method, lumped parameter coupling, electromagnetics AMS subject classifications. 57R19, 58Z05, 65M60, 78M101. Introduction. We present a tool for the homology and cohomology computation of domains tessellated with finite element meshes. The tool is an integrated part of the finite element mesh generator Gmsh [17]. Homology and cohomology computation can be exploited to exhaustively fix the so-called cohomology class of the solution of a boundary value problem that is solved with the finite element method. As a concrete application, we demonstrate how such computations greatly benefit the modeling of an induction heating machine.In boundary value problems that involve the Hodge-Laplace operator, one often needs to choose the cohomology class of the solution. Such problems are usual in electromagnetics, which is why our working example is chosen from that field. The cohomology classes of the problem are generated by the choice of the boundary conditions and by the homology of the problem domain. Informally, homology is about the quantity and the quality of holes in an object, whether it has voids or tunnels or both. Relative homology captures whether the object has holes when one "disregards" a part of the model. In the finite element method, the disregarded part is a subdomain where the solution is fixed by a boundary condition. Cohomology can be characterized by saying that it assigns quantities to these holes. In boundary value problems such assignments fix the cohomology class of the solution. For the technical definitions of homology and cohomology spaces, see appendix A.In the finite element method, typically only a bounded portion of the device and the surrounding space is modeled. The modeling domain may contain holes, and boundary conditions are assigned to confine the fields and couple them with external phenomena outside the domain. Further, the domain can be split into many coupled regions where different approximations and potential formulations are being employed. These modeling aspects give rise to homology and cohomology and their relative forms in numerical models, since one is required to assign source quantities to entities that are absent from the model.For electrical engineers, an evident manifestation of homology and cohomology are Maxwell's equations in their ...

An accurate inertial measurement unit (IMU) is a necessity when considering an inertial navigation system capable of giving reliable position and velocity estimates even for a short period of time. However, even a set of ideal gyroscopes and accelerometers does not imply an ideal IMU if its exact mechanical characteristics (i.e. alignment and position information of each sensor) are not known. In this paper, the standard multi-position calibration method for consumer-grade IMUs using a rate table is enhanced to exploit also the centripetal accelerations caused by the rotation of the table. Thus, the total number of measurements rises, making the method less sensitive to errors and allowing use of more accurate error models. As a result, the accuracy is significantly enhanced, while the required numerical methods are simple and efficient. The proposed method is tested with several IMUs and compared to existing calibration methods.

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This paper presents a calibration method for consumer-grade accelerometers, gyroscopes, and magnetometers. Considering the calibration of consumer-grade sensors, it is essential that no specialized equipment is required to create reference signals. In addition, the less is required from the reference signals, the more suitable the method is on the field. In the proposed method, the novelty in the calibration of the gyroscopes lies in the exploitation of only the known net rotations between the positions in a multi-position calibration. For accelerometers and magnetometers, the innovation is that the direction of reference signals, the gravity and the magnetic field of the Earth, are estimated with calibration parameters. As a consequence, no precise absolute alignment of the sensors is needed in the calibration. The rotations need not be done about a constant axis. In the proposed method, the biases, scale factors, misalignments, and cross-coupling errors for all the sensors as well as hard iron and soft iron effect for magnetometers were modelled. In addition, the drift of the sensors during the calibration was estimated. As a result, all the sensors were calibrated at once to the same frame, exploiting only a cube and a jig and thus, the method is eligible in the field. To estimate the quality of the calibration results, 95 % confidence intervals were calculated for the calibration parameters. Simulations were done to indicate that the calibration method is unbiased.Index Terms-Multi-position calibration, inertial measurement unit (IMU), accelerometer, gyroscope, magnetometer, confidence interval. This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.