Biharmonic Interpolation Research Articles

Sparse sampled visualization of ultrasonic guided waves for defect identification in plate structures

Ultrasonic wave visualization is an effective tool to identify defects along with their location and topology in guided wave structural health monitoring applications. However, conventional imaging techniques require relatively longer data acquisition time and large storage space. Reconstructing waves from sparse sampled images can potentially overcome such difficulties. In this regard, sparse sampling techniques based on random sampling and uniform sampling are explored in this paper. Specifically random sparse sampling using Poisson disk with biharmonic interpolation (PDS-BI) technique is shown to achieve defect identification with an image sparsity of 95% and a sensor density of 5λ2. On the other hand, a novel technique based on uniform sparse sampling with carrier multiplication (US-CM) is demonstrated to reconstruct the image with a sparsity of greater than 98% and a sensor density less than 2λ2. The performance of the proposed techniques has been quantified using Structural Similarity (SSIM) metric and validated through experiments. The reconstructed images are found to be in good agreement with those obtained by a dense array with much larger number of receivers. We also experimentally demonstrate the visualization of ultrasonic wave propagation in a metallic plate using a surface-bonded fiber Bragg grating sensor as an alternative to the conventional sensor.

Selective methods of TOC content estimation for organic-rich interbedded mudstone source rocks

The total organic carbon content (TOC) is one of the key parameters for evaluating the hydrocarbon generation potential of source rocks. Petroleum geochemists and geologists usually use conventional logging curves to predict TOC content in order to reduce the errors caused by limited TOC content data. The ΔlogR method is the most widely used TOC prediction method. However, it is not applicable to organic-rich mudstone interlayers, due to the rapidly changing lithology and the difficulty in determining a baseline. Some improved ΔlogR methods and artificial neural network techniques have also been proposed by previous authors. In this paper, a 3D surface fitting technique based on biharmonic interpolation is proposed to optimize an improved ΔlogR method. A total of 76 samples were divided into pure mudstone and mudstone interlayers. The correlation coefficient (R) and the root mean square error (RMSE) between the measured TOC and the predicted TOC obtained using the improved ΔlogR method, the artificial neural network method, and the 3D surface fitting method were calculated. These methods were applied to assess the source rocks of the Eocene Liushagang Formation in the Fushan Depression, South China Sea. The results show that the 3D surface fitting method can effectively distinguish source rocks from non-source rocks and has a higher accuracy, which makes it the most suitable method for assessing the TOC content of organic-rich source rocks with alternating mudstone layers.

The study of the characteristics of photovoltaic power stations and the identification of rational operating modes by changing the inclination and the spectral composition of the irradiation

The global trends in the development of photovoltaic power stations (PPS) and their application are actively fighting for a place in the sun. On January 20, 2009, Vladimir Putin (at that time the chairman of the government) signed a decree approving the main directions of state policy in the development of renewable energy sources (RES) until 2024, and on May 13, 2019, by decree no. 216 “On the approval of the energy security of the Russian Federation” signed the Doctrine of the energy security of the Russian Federation. In this regard, the need to find the optimal operating modes of PPS has become acute, which is an environmentally friendly way to generate electricity. New designs and methods of using systems are being developed to study and determine the optimal operating modes of PPS. The work justified the design parameters and experimentally investigated the characteristics and operating modes of PPS, determined the effect of the spectral composition of radiation on the amount of generated energy. The study was conducted using graph-analytical processing of the results of practical PPS tests characteristics by simulating outlook surfaces using the biharmonic interpolation method in the Curve Fitting Toolbox application of the MATLAB program.

Visualizing fuzzy sets using opacity-varying freeform diagrams

As an extension of classical sets, fuzzy sets allow their containing elements to have continuous membership, which broadens its applications in various domains. In order to visualize the uncertain owner–member relationship, we design an intuitive diagram to visualize fuzzy sets. The diagram has a freeform boundary, which allows a clear layout of the data. In addition, the opacity of the diagram reveals the uncertainty of the membership. Physical simulation and bi-harmonic interpolation are introduced to generate the proposed diagram. We test our method with different fuzzy sets. The results show that our method is efficient to generate intuitive visualization of fuzzy sets.

Analysis of irradiance losses on a soiled photovoltaic panel using contours

This paper introduces an irradiance loss factor that quantifies the relationship between irradiance, tilt angle and power output of a soiled panel with the soil particle size composition. Artificial soiling experiments were performed using four soil samples at irradiance levels between 200 and 1200W/m2 at 18 tilt angles. Biharmonic interpolation was used to develop power contours in terms of irradiance and tilt angle from experimentally obtained data. These contours were compared with ideal ones of a clean panel to observe deviation in the nature of contours for a soiled panel. A correction factor in terms of particle size composition (as a coefficient to tilt angle) was proposed to calculate power output of a tilted soiled panel. The angular loss on a panel with soil sample containing 150μm particle size in abundance was observed to be 22% and for sample containing 75μm particles in majority, the loss is 24%. Presence of 300μm particle size in abundance causes a 23.7% loss, while 52% angular loss was observed for soil with highest composition of less than 75μm particle size.

Prediction of the lifetime of droplets emitted from mechanical cooling towers by numerical investigation

A numerical modelling capable to simulate the drift and the evaporation of water droplets emitted by a mechanical cooling tower in an urban area is presented. The model is based on a real mechanical draft cooling tower situated in the surroundings of the Miguel Hernández University (Elche, Spain). An experimental study of the deposition of droplets from the cooling tower is done in order to validate the numerical modelling. This study is performed by means of the water sensitive papers technique. A biharmonic interpolation is used for obtaining the total deposition on the floor. A total of 14 cases, everyone with different atmospheric conditions, have been simulated and experimentally validated. An analytical model for the droplets lifetime prediction is validated with the results obtained from the numerical modelling. The present study shows the influence of the atmospheric and droplets conditions in the droplets lifetime, providing useful information to analyse the spread of contaminants or bacteria inside the droplets released from the cooling tower.

Biharmonic optimization of piecewise planar surfaces

In this paper, we propose a method for optimizing geometric objects such as polygons and triangulated piecewise planar surfaces, for the purposes of greater detailing and providing visual smoothness of their images. The initial object is treated as a geometric realization of a simplicial scheme. The method involves refining the simplicial scheme and biharmonic interpolation of the embedding function of a subdivision of the simplicial scheme in Euclidean space.

SU-D-103-07: Dose Measurement and Estimation Methods in An Elliptical Body Phantom for a Conebeam CT System

Purpose: To propose new metrics to estimate a mean dose of an automatic exposure control-enabled angiographic C-arm system over a noncircular large body-shaped phantom based on multi-point dose measurements. Methods: Dose was measured at 9 points in 2 central subregions (C1, C2) and 14 points in 2 peripheral subregions (P1, P2) of the body phantom using a small 0.6cc ion chamber (IC) while operating the system at 16 different combinations of tube voltage, detector dose request, and vertical collimation. In order to acquire complete 2D dose profiles in an axial direction, we carried out Monte Carlo (MC) simulations. After validating the MC model by comparing it to chamber values, the mean dose from MC simulations was used as a ground truth for our mean dose metrics. Mean dose was estimated in 3 ways: 1) Area ratio for each point weights the contribution of the point. 2) Point dose surface fitting method using biharmonic interpolation model. 3) The acquired MC dose profile-based MC template method. We investigated how each method's performance varies as a function of the number of measured data points (7∼23 points). Results: The error of MC compared to chamber readings was 0.9mGy (+/− 0.03 mGy) per point. The relative errors of 1), 2), and 3) methods with 23 IC points in comparison with the MC mean dose were 0.6, 3.37, and 1.9%, respectively. Method 1 performed best for 6 different cases of number of points measured. However, its performance fluctuated compared to methods 2 and 3. Method 3 remained within 3% error with 23∼8 points and showed the most stable performance. Method 2 performed worst with 23∼11 points, with constant error of ∼ 5%. Conclusion: The 3 metrics estimated a mean dose accumulated in a body phantom with about 5% relative error using only 11 points. This work was supported by National Institutes of Health (NIH SIG S10 RR026714-01), by Siemens Medical Solutions, AX, and by the Lucas Foundation. There is no conflict of interest to disclose.

A penalty method for a mixed nonlinear complementarity problem

We propose a power penalty method for a mixed nonlinear complementarity problem (MNCP) and show that the solution to the penalty equation converges to that of the MNCP exponentially as the penalty parameter approaches infinity, provided that the mapping involved in the MNCP is both continuous and ξ -monotone. Furthermore, a convergence theorem is established when the monotonicity assumption on the mapping is removed. To demonstrate the usefulness and the convergence rates of this method, we design a non-trivial test MNCP problem arising in shape-preserving bi-harmonic interpolation and apply our method to this test problem. The numerical results confirm our theoretical findings.

Multi-level meshless methods based on direct multi-elliptic interpolation

A short overview on the direct multi-elliptic interpolation and the related meshless methods for solving partial differential equations is given. A new technique is proposed which produces a biharmonic interpolation along the boundary and solves the original problem inside the domain. An error estimation is also derived. To implement the method, quadtree-based multi-level methods are used. The approach avoids the use of large, dense and ill-conditioned matrices and significantly reduces the computational cost.

Fast Radial Basis Function Interpolation via Preconditioned Krylov Iteration

We consider a preconditioned Krylov subspace iterative algorithm presented by Faul, Goodsell, and Powell (IMA J. Numer. Anal. 25 (2005), pp. 1-24) for computing the coefficients of a radial basis function interpolant over $N$ data points. This preconditioned Krylov iteration has been demonstrated to be extremely robust to the distribution of the points and the iteration rapidly convergent. However, the iterative method has several steps whose computational and memory costs scale as $O(N^{2}),$ both in preliminary computations that compute the preconditioner and in the matrix-vector product involved in each step of the iteration. We effectively accelerate the iterative method to achieve an overall cost of $O(N\log N).$ The matrix vector product is accelerated via the use of the fast multipole method. The preconditioner requires the computation of a set of closest points to each point. We develop an $O(N\log N)$ algorithm for this step as well. Results are presented for multiquadric interpolation in $\mathbb{R}^{2}$ and biharmonic interpolation in $\mathbb{R}^{3}$. A novel FMM algorithm for the evaluation of sums involving multiquadric functions in $\mathbb{R}^{2}$ is presented as well.

Multigrid technique for biharmonic interpolation with application to dual and multiple reciprocity method

A biharmonic-type interpolation method is presented to solve 2D and 3D scattered data interpolation problems. Unlike the methods based on radial basis functions, which produce a large linear system of equations with fully populated and often non-selfadjoint and ill-conditioned matrix, the presented method converts the interpolation problem to the solution of the biharmonic equation supplied with some non-usual boundary conditions at the interpolation points. To solve the biharmonic equation, fast multigrid techniques can be applied which are based on a non-uniform, non-equidistant but Cartesian grid generated by the quadtree/octtree algorithm. The biharmonic interpolation technique is applied to the multiple and dual reciprocity method of the BEM to convert domain integrals to the boundary. This makes it possible to significantly reduce the computational cost of the evaluation of the appearing domain integrals as well as the memory requirement of the procedure. The resulting method can be considered as a special grid-free technique, since it requires no domain discretisation.

Animated images of cardiac membrane voltage during defibrillation

Optical recording using voltage-sensitive dyes has been used to investigate the mechanisms of defibrillation because it (1) is immune to the artifacts produced by high-voltage shocks, (2) provides the time course of the membrane action potential, and (3) can be used to make simultaneous recordings at many sites. The authors used the laser scanning technique to optically record action potentials from 100 sites with 1-ms resolution on the surface of the isolated, perfused rabbit heart during defibrillation. The data were typically analyzed by constructing maps of impulse propagation and examining individual recordings from sites of interest. Described here is a new analysis method that creates millisecond-by-millisecond images of the spatial distribution of membrane potentials. The experimental protocol applied a test shock to the fibrillating heart, followed by a rescue shock and a paced beat. Optical recordings were calibrated to yield membrane voltage as a percentage of the resting and overshoot levels of the postrescue stimulated action potential. The positions of the recording sites and the membrane voltage levels for all 100 sites during a single 1-ms interval were used to interpolate membrane voltage levels at points within a 128 × 128 pixel frame using the biharmonic interpolation method. The level of membrane potential was encoded by pixel color and surface elevation. Sequential frames were viewed as a face-on two-dimensional or as a three-dimensional perspective of the colored surface. Animation of membrane voltage distributions enabled the visualization of the interaction between the shock-induced electrophysiologic response and the propagation of electrical activity preceding and following a defibrillation shock. Successful defibrillation shocks synchronized repolarization across the surface of the heart following the shock.