The finite element methods are important and powerful tools for solving partial differential equations on complex geometries, but their high-order generalizations pose significant challenges in terms of robustness and mesh generation. In this paper, we introduce a high-order finite element method, which is insensitive to element quality and hence is more robust, and which requires only linear elements even for curved geometries and hence significantly simplifies meshing. Based on our recent work on Adaptive Extended Stencil FEM, or AES-FEM (Int. J. Num. Meth. Engrg., 2016, DOI:10.1002 nme.5246), the proposed method replaces the traditional interpolatory FEM basis functions with generalized Lagrange polynomial basis functions, constructed via local weighted least-squares approximations. In this work, we show that AES-FEM can achieve high-order accuracy using linear elements to discretize the geometry and using the standard FEM hat functions as test functions even for curved geometries. This is in contrast to most high-order generalizations of FEM, which require high-order curved elements for curved geometries. We describe the selection of stencils and the treatment of boundary conditions for high-order AES-FEM, and show that even-degree polynomial basis functions are preferred for even-order PDEs. We present numerical results in 2D and 3D for the Poisson equation and the convection-diffusion equation and demonstrate up to sixth order accuracy with AES-FEM. Overall, AES-FEM is much easier to use than FEM for curved boundaries, while being more accurate, more stable, and more efficient in terms of runtime versus error over relatively fine meshes.
The isoparametric FEM in general uses the same-degree polynomial basis uniformly. The @math -adaptive FEM generalizes it by allowing the degrees of the basis functions to vary from element to element @cite_33 @cite_24 . The @math -adaptivity may be further combined with @math -adaptivity, leading to the @math -adaptive FEM or @math -FEM @cite_16 @cite_26 . In these methods, inter-element continuity is typically enforced as constraints @cite_40 . The @math -FEM can achieve high-order convergence, with error decreasing exponentially in the degree of basis functions, provided that the mesh is nearly optimal @cite_2 . Similar to isoparametric FEM, the @math -FEM also requires curved elements for curved geometries, requires good element shapes for accuracy, and is limited to only moderately high orders due to the use of equally spaced nodes.
@cite_26 @cite_33 @cite_24 @cite_40 @cite_2 @cite_16
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We introduce a single unifying framework for a wide range of content-aware image warping tasks using a finite element method (FEM). Existing approaches commonly define error terms over vertex finite differences and can be expressed as a special case of our general FEM model. In this work, we exploit the full generality of FEMs, gaining important advantages over prior methods. These advantages include arbitrary mesh connectivity allowing for adaptive meshing and efficient large-scale solutions, a well-defined continuous problem formulation that enables clear analysis of existing warping error functions and allows us to propose improved ones, and higher order basis functions that allow for smoother warps with fewer degrees of freedom. To support per-element basis functions of varying degree and complex mesh connectivity with hanging nodes, we also introduce a novel use of discontinuous Galerkin FEM. We demonstrate the utility of our method by showing examples in video retargeting and camera stabilization applications, and compare our results with previous state of the art methods. In medical simulations involving tissue deformation, the finite element method (FEM) is a widely used technique, where the size, shape, and placement of the elements in a model are important factors that affect the interpolation and numerical errors of a solution. Conventional model generation schemes for FEM consist of a segmentation step delineating the anatomy followed by a meshing step generating elements conforming to this segmentation. In this paper, a single-step model generation technique is proposed based on optimization. Starting from an initial mesh covering the domain of interest, the mesh nodes are adjusted to minimize an objective function which penalizes intra-element intensity variations and poor element geometry for the entire mesh. Trade-offs between mesh geometry quality and intra-element variance are achieved by adjusting the relative weights of the geometric and intensity variation components of the cost function. This meshing approach enables a more accurate rendering of shapes with fewer elements and provides more accurate models for deformation simulation, especially when the image intensities represent a mechanical feature of the tissue such as the elastic modulus. The use of the proposed mesh optimization is demonstrated in 2-D and 3-D on synthetic phantoms, MR images of the brain, and CT images of the kidney. A comparison with previous meshing techniques that do not account for image intensity is also provided demonstrating the benefits of our approach. The implementation of @math hp-adaptivity is challenging as hanging nodes, edges, and faces have to be constrained to ensure compatibility of the shape functions. For this reason, most @math hp-code frameworks restrict themselves to @math 1-irregular meshes to ease the implementational effort. This work alleviates these difficulties by introducing a new formulation for high-order mesh adaptivity that provides full local @math hp-refinement capabilities at a comparably small implementational effort. Its main idea is the extension of the @math hp- @math d-method such that it allows for high-order overlay meshes yielding a hierarchical, multi-level @math hp-formulation of the Finite Element Method. This concept enables intuitive refinement and coarsening procedures, while linear independence and compatibility of the shape functions are guaranteed by construction. The proposed method is demonstrated to achieve exponential rates of convergence--both in terms of degrees of freedom and in run-time--for problems with non-smooth solutions. Furthermore, the scheme is used alongside the Finite Cell Method to simulate the heat flow around moving objects on a non-conforming background mesh and is combined with an energy-based refinement indicator for automatic @math hp-adaptivity. In this paper we present a new automatic adaptivity algorithm for the hp-FEM which is based on arbitrary-level hanging nodes and local element projections. The method is very simple to implement compared to other existing hp-adaptive strategies, while its performance is comparable or superior. This is demonstrated on several numerical examples which include the L-shape domain problem, a problem with internal layer, and the Girkmann problem of linear elasticity. With appropriate simplifications, the proposed technique can be applied to standard lower-order and spectral finite element methods. The development of numerical methods for solving the Helmholtz equation, which behaves robustly with respect to the wave number, is a topic of vivid research. It was observed that the solution of the Galerkin finite element method (FEM) differs significantly from the best approximation with increasing wave number. Many attempts have been presented in the literature to eliminate this lack of robustness by various modifications of the classical Galerkin FEM. However, we will prove that, in two and more space dimensions, it is impossible to eliminate this so-called pollution effect. Furthermore, we will present a generalized FEM in one dimension which behaves robustly with respect to the wave number. Machine learning (ML) to model the liver biomechanical behaviour during breathing.ML is much faster than the popular FEM, allowing real-time soft tissue modelling.Modelling scheme able to predict deformation for a new load and a new liver.ML regression models were used: three tree-based methods and two simpler ones.Good prediction performance was obtained: all samples with an error under 1źmm. Progress in biomechanical modelling of human soft tissue is the basis for the development of new clinical applications capable of improving the diagnosis and treatment of some diseases (e.g. cancer), as well as the surgical planning and guidance of some interventions. The finite element method (FEM) is one of the most popular techniques used to predict the deformation of the human soft tissue due to its high accuracy. However, FEM has an associated high computational cost, which makes it difficult its integration in real-time computer-aided surgery systems. An alternative for simulating the mechanical behaviour of human organs in real time comes from the use of machine learning (ML) techniques, which are much faster than FEM. This paper assesses the feasibility of ML methods for modelling the biomechanical behaviour of the human liver during the breathing process, which is crucial for guiding surgeons during interventions where it is critical to track this deformation (e.g. some specific kind of biopsies) or for the accurate application of radiotherapy dose to liver tumours. For this purpose, different ML regression models were investigated, including three tree-based methods (decision trees, random forests and extremely randomised trees) and other two simpler regression techniques (dummy model and linear regression). In order to build and validate the ML models, a labelled data set was constructed from modelling the deformation of eight ex-vivo human livers using FEM. The best prediction performance was obtained using extremely randomised trees, with a mean error of 0.07źmm and all the samples with an error under 1źmm. The achieved results lay the foundation for the future development of some real-time software capable of simulating the human liver deformation during the breathing process during clinical interventions. Abstract Objectives Accurate reconstruction and visualisation of soft tissue deformation in real time is crucial in image-guided surgery, particularly in augmented reality (AR) applications. Current deformation models are characterised by a trade-off between accuracy and computational speed. We propose an approach to derive a patient-specific deformation model for brain pathologies by combining the results of pre-computed finite element method (FEM) simulations with machine learning algorithms. The models can be computed instantaneously and offer an accuracy comparable to FEM models. Method A brain tumour is used as the subject of the deformation model. Load-driven FEM simulations are performed on a tetrahedral brain mesh afflicted by a tumour. Forces of varying magnitudes, positions, and inclination angles are applied onto the brain's surface. Two machine learning algorithms—artificial neural networks (ANNs) and support vector regression (SVR)—are employed to derive a model that can predict the resulting deformation for each node in the tumour's mesh. Results The tumour deformation can be predicted in real time given relevant information about the geometry of the anatomy and the load, all of which can be measured instantly during a surgical operation. The models can predict the position of the nodes with errors below 0.3 mm, beyond the general threshold of surgical accuracy and suitable for high fidelity AR systems. The SVR models perform better than the ANN's, with positional errors for SVR models reaching under 0.2 mm. Conclusions The results represent an improvement over existing deformation models for real time applications, providing smaller errors and high patient-specificity. The proposed approach addresses the current needs of image-guided surgical systems and has the potential to be employed to model the deformation of any type of soft tissue. We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . We present a new method to convert the geometry of a mesh into frequency space. The eigenfunctions of the Laplace-Beltrami operator are used to define Fourier-like function basis and transform. Since this generalizes the classical Spherical Harmonics to arbitrary manifolds, the basis functions will be called Manifold Harmonics. It is well known that the eigenvectors of the discrete Laplacian define such a function basis. However, important theoretical and practical problems hinder us from using this idea directly. From the theoretical point of view, the combinatorial graph Laplacian does not take the geometry into account. The discrete Laplacian (cotan weights) does not have this limitation, but its eigenvectors are not orthogonal. From the practical point of view, computing even just a few eigenvectors is currently impossible for meshes with more than a few thousand vertices. In this paper, we address both issues. On the theoretical side, we show how the FEM (Finite Element Modeling) formulation defines a function basis which is both geometry-aware and orthogonal. On the practical side, we propose a band-by-band spectrum computation algorithm and an out-of-core implementation that can compute thousands of eigenvectors for meshes with up to a million vertices. Finally, we demonstrate some applications of our method to interactive convolution geometry filtering and interactive shading design. Abstract The present paper describes a method to enhance the capability of, or to broaden the scope of computational mechanics by using deep learning, which is one of the machine learning methods and is based on the artificial neural network. The method utilizes deep learning to extract rules inherent in a computational mechanics application, which usually are implicit and sometimes too complicated to grasp from the large amount of available data A new method of numerical quadrature for the FEM stiffness matrices is developed by using the proposed method, where a kind of optimized quadrature rule superior in accuracy to the standard Gauss–Legendre quadrature is obtained on the element-by-element basis. The detailed formulation of the proposed method is given with the sample application above, and an acceleration technique for the proposed method is discussed This chapter presents new real-time and physics-based modeling methods dedicated to deformable soft robots. In this approach, continuum mechanics provides the partial derivative equations that govern the deformations, and Finite Element Method (FEM) is used to compute numerical solutions adapted to the robot. A formulation based on Lagrange Multipliers is used to model the behavior of the actuators as well as the contact with the environment. Direct and inverse kinematic models are also obtained for real-time control. Some experiments and numerical results are presented. In this paper, we present a new method for the control of soft robots with elastic behavior, piloted by several actuators. The central contribution of this work is the use of the Finite Element Method (FEM), computed in real-time, in the control algorithm. The FEM based simulation computes the nonlinear deformations of the robots at interactive rates. The model is completed by Lagrange multipliers at the actuation zones and at the end-effector position. A reduced compliance matrix is built in order to deal with the necessary inversion of the model. Then, an iterative algorithm uses this compliance matrix to find the contribution of the actuators (force and or position) that will deform the structure so that the terminal end of the robot follows a given position. Additional constraints, like rigid or deformable obstacles, or the internal characteristics of the actuators are integrated in the control algorithm. We illustrate our method using simulated examples of both serial and parallel structures and we validate it on a real 3D soft robot made of silicone. We describe a new explicit function that given an elliptic curve E defined over @math , maps elements of @math into E in deterministic polynomial time and in a constant number of operations over @math . The function requires to compute a cube root. As an application we show how to hash deterministically into an elliptic curve. We consider the problem of finding a minimum edge cost subgraph of a graph satisfying both given node-connectivity requirements and degree upper bounds on nodes. We present an iterative rounding algorithm of the biset linear programming relaxation for this problem. For directed graphs and @math -out-connectivity requirements from a root, our algorithm computes a solution that is a 2-approximation on the cost, and the degree of each node @math in the solution is at most @math , where @math is the degree upper bound on @math . For undirected graphs and element-connectivity requirements with maximum connectivity requirement @math , our algorithm computes a solution that is a @math -approximation on the cost, and the degree of each node @math in the solution is at most @math . These ratios improve the previous @math -approximation on the cost and @math -approximation on the degrees. Our algorithms can be used to improve approximation ratios for other node-connectivity problems such as undirected $k... Finite Element analysis can provide accurate deformable models for soft-robots. However, using such models is very difficult in a real-time system of control. In this paper, we introduce a generic solution that enables a high-rate control and that is compatible with strong real-time constraints. From a Finite Element analysis, computed at low rate, an inverse model of the robot outputs the setpoint values for the actuator in order to obtain a desired trajectory. This inverse problem uses a QP (quadratic-programming) algorithm based on the equations set by the Finite Element Method. To improve the update rate performances, we propose an asynchronous simulation framework that provides a better trade-off between the deformation accuracy and the computational burden. Complex computations such as accurate FEM deformations are done at low frequency while the control is performed at high frequency with strong real-time constraints. The two simulation loops (high frequency and low frequency loops) are mechanically coupled in order to guarantee mechanical accuracy of the system over time. Finally, the validity of the multi-rate simulation is discussed based on measurements of the evolution in the QP matrix and an experimental validation is conducted to validate the correctness of the high-rate inverse model on a real robot. Since it relies on a geometrical rather than a variational framework, many find the finite volume method (FVM) more intuitive than the finite element method (FEM). We show that the FVM allows one to interpret the stress inside a tetrahedron as a simple "multidimensional force" pushing on each face. Moreover, this interpretation leads to a heuristic method for calculating the force on each node, which is as simple to implement and comprehend as masses and springs. In the finite volume spirit, we also present a geometric rather than interpolating function definition of strain. We use the FVM and a quasi-incompressible, transversely isotropic, hyperelastic constitutive model to simulate contracting muscle tissue. B-spline solids are used to model fiber directions, and the muscle activation levels are derived from key frame animations. We develop fast and memory efficient numerical methods for learning functions of many variables that admit sparse representations in terms of general bounded orthonormal tensor product bases. Such functions appear in many applications including, e.g., various Uncertainty Quantification(UQ) problems involving the solution of parametric PDE that are approximately sparse in Chebyshev or Legendre product bases. We expect that our results provide a starting point for a new line of research on sublinear-time solution techniques for UQ applications of the type above which will eventually be able to scale to significantly higher-dimensional problems than what are currently computationally feasible. More concretely, let @math be a finite Bounded Orthonormal Product Basis (BOPB) of cardinality @math . We will develop methods that approximate any function @math that is sparse in the BOPB, that is, @math of the form @math with @math of cardinality @math . Our method has a runtime of just @math , uses only @math function evaluations on a fixed and nonadaptive grid, and not more than @math bits of memory. For @math , the runtime @math will be less than what is required to simply enumerate the elements of the basis @math ; thus our method is the first approach applicable in a general BOPB framework that falls into the class referred to as "sublinear-time". This and the similarly reduced sample and memory requirements set our algorithm apart from previous works based on standard compressive sensing algorithms such as basis pursuit which typically store and utilize full intermediate basis representations of size @math . Given data stream @math of size @math of numbers from @math , the frequency of @math is defined as @math . The @math -th of @math is defined as @math . We consider the problem of approximating frequency moments in insertion-only streams for @math . For any constant @math we show an @math upper bound on the space complexity of the problem. Here @math is the iterative @math function. To simplify the presentation, we make the following assumptions: @math and @math are polynomially far; approximation error @math and parameter @math are constants. We observe a natural bijection between streams and special matrices. Our main technical contribution is a non-uniform sampling method on matrices. We call our method a ; it samples a heavy element (i.e., element @math with frequency @math ) with probability @math and gives approximation @math . In addition, the estimations never exceed the real values, that is @math for all @math . As a result, we reduce the space complexity of finding a heavy element to @math bits. We apply our method of recursive sketches and resolve the problem with @math bits. We consider range queries that search for low-frequency elements (least frequent elements and @math ?-minorities) in arrays. An @math ?-minority of a query range has multiplicity no greater than an @math ? fraction of the elements in the range. Our data structure for the least frequent element range query problem requires @math O(n) space, @math O(n3 2) preprocessing time, and @math O(n) query time. A reduction from boolean matrix multiplication to this problem shows the hardness of simultaneous improvements in both preprocessing time and query time. Our data structure for the @math ?-minority range query problem requires @math O(n) space, supports queries in @math O(1 ?) time, and allows @math ? to be specified at query time. This paper studies problems of inferring order given noisy information. In these problems there is an unknown order (permutation) @math on @math elements denoted by @math . We assume that information is generated in a way correlated with @math . The goal is to find a maximum likelihood @math given the information observed. We will consider two different types of observations: noisy comparisons and noisy orders. The data in Noisy orders are permutations given from an exponential distribution correlated with (this is also called the Mallow's model). The data in Noisy Comparisons is a signal given for each pair of elements which is correlated with their true ordering. In this paper we present polynomial time algorithms for solving both problems with high probability. As part of our proof we show that for both models the maximum likelihood solution @math is close to the original permutation @math . Our results are of interest in applications to ranking, such as ranking in sports, or ranking of search items based on comparisons by experts.