Displacement Interpolation Research Articles

Continual Learning of Generative Models With Limited Data: From Wasserstein-1 Barycenter to Adaptive Coalescence.

Learning generative models is challenging for a network edge node with limited data and computing power. Since tasks in similar environments share a model similarity, it is plausible to leverage pretrained generative models from other edge nodes. Appealing to optimal transport theory tailored toward Wasserstein-1 generative adversarial networks (WGANs), this study aims to develop a framework that systematically optimizes continual learning of generative models using local data at the edge node while exploiting adaptive coalescence of pretrained generative models. Specifically, by treating the knowledge transfer from other nodes as Wasserstein balls centered around their pretrained models, continual learning of generative models is cast as a constrained optimization problem, which is further reduced to a Wasserstein-1 barycenter problem. A two-stage approach is devised accordingly: 1) the barycenters among the pretrained models are computed offline, where displacement interpolation is used as the theoretic foundation for finding adaptive barycenters via a "recursive" WGAN configuration and 2) the barycenter computed offline is used as metamodel initialization for continual learning, and then, fast adaptation is carried out to find the generative model using the local samples at the target edge node. Finally, a weight ternarization method, based on joint optimization of weights and threshold for quantization, is developed to compress the generative model further. Extensive experimental studies corroborate the effectiveness of the proposed framework.

Isometric rigidity of Wasserstein spaces over Euclidean spheres

We study the structure of isometries of the quadratic Wasserstein space W2(Sn,‖⋅‖) over the sphere endowed with the distance inherited from the norm of Rn+1. We prove that W2(Sn,‖⋅‖) is isometrically rigid, meaning that its isometry group is isomorphic to that of (Sn,‖⋅‖). This is in striking contrast to the non-rigidity of its ambient space W2(Rn+1,‖⋅‖) but in line with the rigidity of the geodesic space W2(Sn,∢). One of the key steps of the proof is the use of mean squared error functions to mimic displacement interpolation in W2(Sn,‖⋅‖). A major difficulty in proving rigidity for quadratic Wasserstein spaces is that one cannot use the Wasserstein potential technique. To illustrate its general power, we use it to prove isometric rigidity of Wp(S1,‖⋅‖) for 1≤p<2.

PaIRS-UniNa: A Robust And Accurate Free Tool For Digital Particle Image Velocimetry And Optical Camera Calibration

This contribution presents PaIRS-UniNa (PArticle Image Reconstruction Software – University of Naples “Federico II”), a free software application designed to perform digital particle image velocimetry (PIV). PaIRS-UniNa relies on a powerful C library designed for speed, robustness and accuracy, and a user-friendly graphical interface (Figure 1) running in Python environment on all operating systems, including Windows, Linux and MacOS. At the current stage, it features modules to perform two-component (2C) PIV, stereoscopic PIV and optical camera calibration for multi-camera systems. The processing of PIV and stereoscopic PIV data is based on an iterative image deformation method, which can be customized by the user varying different parameters in such a way to find the best compromise between accuracy and computational speed. These parameters include the size, shape and number of interrogation windows, the number of process iterations and the type of interpolating functions for both image deformation and displacement interpolation (such as bilinear, bicubic, methods based on simplexes or the FFT shift theorem and B-splines of varying orders). Additionally, it is possible to select a range of weighting windows for cross-correlation estimation and velocity field filtering, including rectangular, Blackman, Gaussian, Harris, Hann windows, among others, and several vector validation strategies for outlier detection and replacement. Such a level of customization enables to effectively adjust the modulation transfer function of the PIV process to meet specific application requirements. For the stereoscopic PIV process two different approaches are possible: mapping, which involves applying PIV to de-warped images for 2D-2C displacement fields, and warping, which applies PIV to raw images and combine 3C reconstruction and image dewarping in a single subsequent step. In order to accurately estimate the laser sheet plane location and correct errors due to misalignment and inaccurate displacement vector positions, a disparity correction procedure is also available. The calibration module (CalVi) supports accurate calibration with the camera models mostly used in the PIV community: polynomials, rational functions and the pinhole camera model. Moreover, it supports the integration of the pinhole camera model with a refractive correction model for cylindrical geometries and camera calibration procedures working with unknown positions and orientations of the calibration target. In such a way, the target may be moved by hand without losing accuracy in comparison with a conventional calibration, in which the target is displaced with the aid of translational and rotational stages and knowledge of the target positions is strictly required. Already used in numerous experimental studies by the Experimental Thermo-Fluid Dynamics group of University of Naples “Federico II” and international collaborating groups, PaIRS-UniNa offers a rapid and effective free alternative to commercial PIV software for research and industrial use.

Model order reduction by convex displacement interpolation

We present a nonlinear interpolation technique for parametric fields that exploits optimal transportation of coherent structures of the solution to achieve accurate performance. The approach generalizes the nonlinear interpolation procedure introduced in [Iollo, Taddei, J. Comput. Phys., 2022] to multi-dimensional parameter domains and to datasets of several snapshots. Given a library of high-fidelity simulations, we rely on a scalar testing function and on a point set registration method to identify coherent structures of the solution field in the form of sorted point clouds. Given a new parameter value, we exploit a regression method to predict the new point cloud; then, we resort to a boundary-aware registration technique to define bijective mappings that deform the new point cloud into the point clouds of the neighboring elements of the dataset, while preserving the boundary of the domain; finally, we define the estimate as a weighted combination of modes obtained by composing the neighboring snapshots with the previously-built mappings. We present several numerical examples for compressible and incompressible, viscous and inviscid flows to demonstrate the accuracy of the method. Furthermore, we employ the nonlinear interpolation procedure to augment the dataset of simulations for linear-subspace projection-based model reduction: our data augmentation procedure is designed to reduce offline costs — which are dominated by snapshot generation — of model reduction techniques for nonlinear advection-dominated problems.

A robust radial point interpolation method empowered with neural network solvers (RPIM-NNS) for nonlinear solid mechanics

In this work, we proposed a robust radial point interpolation method empowered with neural network solvers (RPIM-NNS) for solving highly nonlinear solid mechanics problems. It is enabled by neural network solvers via minimizing an energy-based functional loss. The RPIM-NNS has the following key ingredients: (1) It uses radial basis functions (RBFs) for displacement interpolation at arbitrary points in the problem domain, permitting irregular node distributions. (2) Nodes are placed also beyond the domain boundary, allowing the convenient implementation of boundary conditions of both Dirichlet and Neumann types. (3) It uses strain energy in an integral form as a part of the loss function, ensuring solution stability. (4) A well-developed gradient descendant algorithm in machine learning is employed to find the optimal solution, enabling robustness and ease in handling material and geometrical nonlinearities. (5) The proposed RPIM-NNS is compatible with parallel computing schemes. The performance of this method is tested using nonlinear problems including Cook's membrane and 3D twisting rubber problems, demonstrating its remarkable stability and robustness. This work, which seamlessly integrates the neural network solvers with mechanics governing equations and computational mechanics techniques, offers an excellent alternative for nonlinear solid mechanics problems. MATLAB codes are made available at https://github.com/JinshuaiBai/RPIM_NNS for free downloading.

Detecting Weak Distribution Shifts via Displacement Interpolation

Detecting weak, systematic distribution shifts and quantitatively modeling individual, heterogeneous responses to policies or incentives have found increasing empirical applications in social and economic sciences. Given two probability distributions P (null) and Q (alternative), we study the problem of detecting weak distribution shift deviating from the null P toward the alternative Q, where the level of deviation vanishes as a function of n, the sample size. We propose a model for weak distribution shifts via displacement interpolation between P and Q, drawing from the optimal transport theory. We study a hypothesis testing procedure based on the Wasserstein distance, derive sharp conditions under which detection is possible, and provide the exact characterization of the asymptotic Type I and Type II errors at the detection boundary using empirical processes. We demonstrate how the proposed testing procedure works in modeling and detecting weak distribution shifts in real datasets using two empirical examples: distribution shifts in consumer spending after COVID-19, and heterogeneity in the published p-values of statistical tests in journals across different disciplines.

Optimal Transport Approaches for Cloud Fusion in 3D Reconstruction

Cloud fusion plays a crucial role in achieving accurate 3D models. This paper introduces two optimal transport-based approaches, Gaussian Fusion and Cyclical Fusion, to showcase the effectiveness of optimal transport in establishing dense correspondence among incoming 3D point clouds. These approaches encompass several important components. Firstly, the fusion correspondences are derived from the optimal transport problem. The core principle of Gaussian Fusion and Cyclical Fusion involves manipulating points along geodesic curves to effectively leverage significant local geometric information. Secondly, this paper explores the fundamental concepts underlying Gaussian Fusion (displacement interpolation) and Cyclical Fusion (cyclical monotonicity), which greatly enhance the accuracy and completeness of the reconstruction process. And provides evidence for the uniqueness of displacement interpolation as geodesics on L2-Wasserstein space and the rationality of cyclical monotonicity in this context. Finally, the proposed approaches excel in capturing intricate surface details, particularly on small objects, when compared to the original fusion scheme which often introduces severe artifacts.

Hexahedral Solid Element with Rotational Degrees of Freedom Based on a Novel Trail-Correction Displacement Interpolation Scheme

Hexahedral Solid Element with Rotational Degrees of Freedom Based on a Novel Trail-Correction Displacement Interpolation Scheme

Rational generalized mixed finite element method for 2D linear elastic problems

Combining the ideas of rational displacement finite element and generalized mixed finite element, the rational generalized mixed element with 5 nodes is proposed. The finite formulations for the proposed element are derived on the basis of the generalized mixed variational principle, and the two field variables of displacement and stress can be solved simultaneously. Fundamental solutions of planar linear elasticity are adopted as the displacement interpolation function, while polynomials satisfying the equilibrium equation are adopted as the stress interpolation function. Both the displacement and stress boundary conditions can be introduced directly at same time in the proposed method, which contributes to being a more rational model. The solution technique is utilized to analyze a couple of representative static problems, and the results show the properties of excellent accuracy and good convergence. In addition, it is found to be evident for improvement of the accuracy when the stress boundary conditions are applied.

Characterization of the null energy condition via displacement convexity of entropy

AbstractWe characterize the null energy condition for an ‐dimensional, time‐oriented Lorentzian manifold in terms of convexity of the relative ‐Renyi entropy along displacement interpolations on null hypersurfaces. More generally, we also consider Lorentzian manifolds with a smooth weight function and introduce the Bakry–Emery ‐null energy condition that we characterize in terms of null displacement convexity of the relative ‐Renyi entropy. As application we then revisit Hawking's area monotonicity theorem for a black hole horizon and the Penrose Singularity Theorem from the viewpoint of this characterization and in the context of weighted Lorentzian manifolds.

LADI: Landslide displacement interpolation through a spatial-temporal Kalman filter

Monitoring landslide displacement is critical towards understanding kinematics and evaluating risk. Both remote sensing technology and in-situ sensors are currently used for monitoring purposes, each with their own spatio-temporal advantages. However, the majority of current workflows analyze these data independently, missing a synergistic opportunity to combine complementary datasets to enhance analyses both spatially and temporally. This work presents the first landslide specific displacement interpolation technique using a novel approach, LADI (Landslide Displacement Interpolation), which creates a high-spatial, high-temporal resolution interpolation of landslide surface displacement by combining remotely-sensed observation data with and in-situ sensor data. At a minimum, LADI requires only a single in-situ monitoring station (control point) and a single high-resolution displacement dataset (calibration data) to function. LADI implements a series of Kalman filters, where interpolation is first performed spatially, then optionally in the temporal domain. LADI is demonstrated on both simulated and real-world landslide data to showcase the behavior, effectiveness, and applicability of the method. Results are compared to common spatial interpolation methods, which demonstrate that LADI achieves superior performance (average EPE = 0.018 m and 0.031 m of LADI and the best performing common spatial interpolation method respectively when number of interpolation control points = 4) in the real-world test sparse displacement dataset when the number of spatial control points is less than sixteen. Additionally, the performance gain of utilizing LADI is further enhanced over common interpolation methods when the interpolation control points are not equally distributed geographically (average EPE = 0.016 m and 0.049 m respectively) as is usually the case with in-situ landslide displacement monitoring stations.

Distributed Wasserstein Barycenters via Displacement Interpolation

Consider a multi-agent system whereby each agent has an initial probability measure. In this paper, we propose a distributed algorithm based upon stochastic, asynchronous and pairwise exchange of information and displacement interpolation in the Wasserstein space. We characterize the evolution of this algorithm and prove it computes the Wasserstein barycenter of the initial measures under various conditions. One version of the algorithm computes a standard Wasserstein barycenter, i.e., a barycenter based upon equal weights; and the other version computes a randomized Wasserstein barycenter, i.e., a barycenter based upon random weights for the initial measures. Finally, we specialize our results to Gaussian distributions and draw a connection with the modeling of opinion dynamics in mathematical sociology.

Thermoelectric Joule heating in crack problems in nano-sized structures

Crack problems in nano-sized elastic structures under the Joule heating are analyzed by the finite element method (FEM). The variational principle is applied for derivation of coupled electrothermomechanical governing equations. The temperature field is described by classical Fourier′s law and it is not influenced by mechanical field. Transient heat conduction is considered since it is the slowest process among all other coupled fields here. Quasi-static conditions are considered for mechanical and electrical fields. Due to higher derivatives in governing equations the conventional FEM cannot be applied. Therefore, the mixed FEM with the C0 continuous interpolation for displacements and strains is developed. The influence of the material characteristic length on variation of crack opening displacements in finite nano-sized structures under an electric current load is investigated.

Two-scale modeling of particle crack initiation and propagation in particle-reinforced composites by using microscale analysis based on Voronoi cell finite element model

In this study, a two-scale finite element model has been developed for the elastic analysis of composite materials. The coarse finite element mesh was assumed for the macroscale analysis, while Voronoi cell finite element model (VCFEM) was used for the microscale analysis of heterogeneous domains. The adaptive coupling between the two scales in non-critical and critical regions was accomplished through the master–slave nodes method. Two scale elements were connected together through the use of shared nodes and the order of the edge displacement interpolation. A new extended VCFEM was proposed to simulate particle damage generation, and a remeshing algorithm was constructed to simulate particle crack generation and propagation in particle-reinforced composites. For crack propagation analysis, four new combined Voronoi elements were developed and analyzed at the microscale. The obtained results for the stresses, stress intensity factors, and crack path were compared with those calculated using the commercial calculation software Abaqus. The two sets of results are in very good agreement, confirming the accuracy of the proposed method used for determining the stress concentrations and also for coarse-mesh discretization.

The Crustal Dynamics and Its Geological Explanation of the Three-Dimensional Co-Seismic Deformation Field for the 2021 Maduo MS7.4 Earthquake Based on GNSS and InSAR.

Three-dimensional deformation is an important input to explore seismic mechanisms and geodynamics. The GNSS and InSAR technologies are commonly used to obtain the co-seismic three-dimensional deformation field. This paper focused on the effect of calculation accuracy caused by the deformation correlation between the reference point and the points involved in the solution, to build a high-accuracy three-dimensional deformation field for a detailed geological explanation. Based on the variance component estimation (VCE) method, the InSAR LOS, azimuthal deformation, and the GNSS horizontal and vertical deformation were integrated to solve the three-dimensional displacement of the study area in combination with the elasticity theory. The accuracy of the three-dimensional co-seismic deformation field of the 2021 Maduo MS7.4 earthquake obtained by the method proposed in this paper, was compared with that obtained from the only InSAR measurements obtained using a multi-satellite and multi-technology approach. The results showed the difference in root-mean-square errors (RMSE) of the integration and GNSS displacement was 0.98 cm, 5.64 cm, and 1.37 cm in the east-west, north-south and vertical direction respectively, which was better than the RMSE of the method using only InSAR and GNSS displacement, which was 5.2 cm and 12.2 cm in the east-west, north-south, and no vertical direction. With the geological field survey and aftershocks relocation, the results showed good agreement with the strike and the position of the surface rupture. The maximum slip displacement was about 4 m, which was consistent with the result of the empirical statistical formula. It was firstly found that the pre-existing fault controlled the vertical deformation on the south side of the west end of the main surface rupture caused by the Maduo MS7.4 earthquake, which provided the direct evidence for the theoretical hypothesis that large earthquakes could not only produce surface rupture on seismogenic faults, but also trigger pre-existing faults or new faults to produce surface rupture or weak deformation in areas far from seismogenic faults. An adaptive method was proposed in GNSS and InSAR integration, which could take into account the correlation distance and the efficiency of homogeneous point selection. Meanwhile, deformation information of the decoherent region could be recovered without interpolation of the GNSS displacement. This series of findings formed an essential supplement to the field surface rupture survey and provided a novel idea for the combination of the various spatial measurement technologies to improve the seismic deformation monitoring.

Characterization of Gromov-type geodesics

It is well known that, when endowed with the Gromov-Hausdorff distance dGH, the collection M of all isometry classes of compact metric spaces is a complete and separable space. It is also known that (M,dGH) is a geodesic metric space, but there is no known structural characterization of geodesics in M.In this paper we provide two characterizations of geodesics in M. We call a Gromov-Hausdorff geodesic γ:[0,1]→MHausdorff-realizable if there exists a compact metric space Z containing isometric copies of γ(t) for each t∈[0,1] such that the Hausdorff distance satisfies dHZ(γ(s),γ(t))=dGH(γ(s),γ(t)) for all s,t∈[0,1]. In this way, γ is actually a geodesic in the Hausdorff hyperspace of Z, and we call it a Hausdorff geodesic. We prove that in fact every Gromov-Hausdorff geodesic is Hausdorff-realizable. Inspired by this characterization, we further elucidate a structural connection between Hausdorff geodesics and Wasserstein geodesics: we show that every Hausdorff geodesic is equivalent to a so-called Hausdorff displacement interpolation. This equivalence allows us to establish that every Gromov-Hausdorff geodesic is dynamic, a notion which we develop in analogy with dynamic optimal couplings in the theory of optimal transport.Besides geodesics in M, we also study geodesics on the collection Mw of isomorphism classes of compact metric measure spaces. Sturm constructed a family of Gromov-type distances on Mw, which we denote dGW,pS (for p∈[1,∞)), as an analogue of dGH, and proved that (Mw,dGW,pS) is also a geodesic space. We define a notion of Wasserstein-realizabledGW,pS geodesics in a sense similar to Hausdorff-realizable geodesics and show that the set of all Wasserstein-realizabledGW,pS geodesics is dense in the set of all dGW,pS geodesics. We further identify a rich class of dGW,pS geodesics which are Wasserstein-realizable.

Generalized Computation of Buckling Loads via Curvature-Based Displacement Interpolation

The calculation of flexural buckling loads for structural members remains a relevant topic of practical and theoretical interest. In addition to well-known solutions for prismatic members, solutions are available for the buckling loads of nonprismatic and weakened members; however, these solutions are often specialized and difficult to implement, as are finite element analyses that require a refined mesh in order to achieve accurate solutions. A straightforward approach for computing buckling loads based on eigenvalue analysis of the curvature-based displacement interpolation (CBDI) influence matrix is developed. The CBDI influence matrix, which is a byproduct of a force-based frame element formulation of geometric nonlinearity, simplifies the calculation of flexural buckling loads for nonprismatic and weakened members while also providing accurate results for prismatic members. Comparisons with previously published solutions show the CBDI approach gives accurate first-mode buckling loads for prismatic and nonprismatic columns when the CBDI influence matrix is formed using at least three interpolation points. More interpolation points are required for the critical buckling loads of columns where the change in flexural stiffness is more abrupt. The CBDI approach is easy to implement and provides engineers and researchers a means of calculating flexural buckling loads for members with arbitrary changes in flexural stiffness.

Finite element formulation for distortional lateral buckling of I-beams

The present study develops a finite element formulation for the distortional buckling of I- beams. The formulation characterizes the distribution of the lateral displacement along the web height by superposing (a) two linear modes intended to capture the classical non-distortional lateral-torsional behaviour and (b) any number of user-specified Fourier terms intended to capture additional web distortion. All displacement fields characterizing the lateral displacements are taken to follow a cubic distribution in the longitudinal direction. The separation of variables is effectively achieved by exploiting the properties of the matrix Kronecker product. The finite element solution developed is shown to replicate accurately (a) the classical non-distortional lateral-torsional buckling solutions (b) previously developed distortional buckling solutions based on cubic interpolation of the lateral displacement, while (c) providing a basis to assess the effect to commonly omitted higher distortional modes on the predicted critical moments and buckling modes. The solution is then used to conduct a systematic parametric study of over 3900 cases to quantify the reduction in critical moments due to web distortion relative to the classical non-distortional predictions in the case of simply supported beams under uniform loads, point loads, and linear moment gradients, cantilevers, and beams with overhang.

Two-scale Modeling of Composites damage with Voronoi cell finite element method for microscale computation

This paper develops an adaptive two-level computational model that includes macro and microscope analysis. Traditional displacement based finite element is used for macroscopic analysis, while Voronoi cell finite element model is used for computations of arbitrary heterogeneous domains. For an adaptive analysis, when local area may be destructive, VCFEM mesh is used instead of corresponding finite element for damage analysis. Because VCFEM model and finite element are all taken the nodal displacements as field variables, therefore, based on the idea of multigrid by the adaptive mesh adjustment, let finite element edge length and VCFEM edge length are basically the same, two kinds of elements were connected together through the use of shared nodes and use the order of the edge displacement interpolation. This formed a two-scale model of the coexistence of macro structure and micro structure. For microscale analysis, three Voronoi cell finite models (VCFEM) are build including perfect interface bonding model, interface debonding model and multiple matrix cracks model. Numerical example is executed for validating the effectiveness of the two-scale model method.

Hellinger–Reissner principle based stress–displacement formulation for three-dimensional isogeometric analysis in linear elasticity

In case of Lagrangian finite element formulation, three-dimensional (3-D) stress-based hybrid solid elements have shown excellent coarse mesh accuracy for a wide range of applications. However, to the best of our knowledge, there is no work available towards the development of 3-D stress-based hybrid solid elements for isogeometric analysis (IGA). In this work, we propose stress-based hybrid solid elements to alleviate the issue of locking arising in non-uniform rational B-spline (NURBS)-based IGA. The present work primarily focuses on the linear elasticity, though the formulation can be extended to the non-linear regime. We believe that the excellent coarse mesh accuracy provided by the proposed elements will further enhance the IGA in various applications, especially involving structures with high aspect ratios and nearly incompressible materials. The proposed elements are constructed based on a two-field Hellinger–Reissner variational statement, where stress and displacement fields are interpolated separately. The stress interpolation functions have been derived systemically for various orders of NURBS displacement interpolation functions. Further, we show mathematically that the choice of the stress interpolation functions is free from spurious zero-energy modes. Lastly, the results of numerous 3-D linear-elastic benchmark problems are presented to demonstrate the efficacy and robustness of the proposed elements. The results confirm the superior coarse mesh accuracy for structures with high aspect ratios and almost incompressible materials.