Nonlinear Energy Function Research Articles

Signorini problem as a variational limit of obstacle problems in nonlinear elasticity

<abstract><p>An energy functional for the obstacle problem in linear elasticity is obtained as a variational limit of nonlinear elastic energy functionals describing a material body subject to pure traction load under a unilateral constraint representing the rigid obstacle. There exist loads pushing the body against the obstacle, but unfit for the geometry of the whole system body-obstacle, so that the corresponding variational limit turns out to be different from the classical Signorini problem in linear elasticity. However, if the force field acting on the body fulfils an appropriate geometric admissibility condition, we can show coincidence of minima. The analysis developed here provides a rigorous variational justification of the Signorini problem in linear elasticity, together with an accurate analysis of the unilateral constraint.</p></abstract>

Preprint: Global, and local optimization beamforming for broadband sources.

This paper presents an alternative energy function for global optimization (GO) beamforming, tailored to acoustic broadband sources. Given that properties, such as the source location, multipole rotation, or flow conditions, are parameterized over the frequency, a cross spectral matrix-fitting can be performed for all frequencies at once. A numerical analysis shows that the nonlinear energy function for the standard GO problem is equivalent to the source's point spread function (PSF) and contains local minima at the locations of the grating lobes and side lobes. The energy function is improved with the proposed broadband energy, as it averages the PSF. Further, it simplifies the process of identifying sources and reconstructing their spectra from the results. The paper shows that the method is superior on synthetic monopoles compared to standard GO and CLEAN based on source coherence (CLEAN-SC). For real-world data, the results of the proposed method and CLEAN-SC are similar, and outperform standard GO. The main difference is that source assumption violations cause noisy maps for CLEAN-SC and cause wrong spectral estimations of the proposed method. By using reasonable initial values, the GO problem reduces to a local optimization problem with similar results. Further, the proposed method is able to identify synthetic multipoles with different pole amplitudes and unknown pole rotations.

Discrete nonlinear elastodynamics in a port‐Hamiltonian framework

AbstractWe provide a fully nonlinear port‐Hamiltonian formulation for discrete elastodynamical systems as well as a structure‐preserving time discretization. The governing equations are obtained in a variational manner and represent index‐1 differential algebraic equations. Performing an index reduction, one obtains the port‐Hamiltonian state space model, which features the nonlinear strains as an independent state next to position and velocity. Moreover, hyperelastic material behavior is captured in terms of a nonlinear stored energy function. The model exhibits passivity and losslessness and has an underlying symmetry yielding the conservation of angular momentum. We perform temporal discretization using the midpoint discrete gradient, such that the beneficial properties are inherited by the developed time stepping scheme in a discrete sense. The numerical results obtained in a representative example are demonstrated to validate the findings.

As-locally-uniform-as-possible reshaping of vector clip-art

Vector clip-art images consist of regions bounded by a network of vector curves. Users often wish to reshape , or rescale, existing clip-art images by changing the locations, proportions, or scales of different image elements. When reshaping images depicting synthetic content they seek to preserve global and local structures. These structures are best preserved when the gradient of the mapping between the original and the reshaped curve networks is locally as close as possible to a uniform scale; mappings that satisfy this property maximally preserve the input curve orientations and minimally change the shape of the input's geometric details, while allowing changes in the relative scales of the different features. The expectation of approximate scale uniformity is local ; while reshaping operations are typically expected to change the relative proportions of a subset of network regions, users expect the change to be minimal away from the directly impacted regions and expect such changes to be gradual and distributed as evenly as possible. Unfortunately, existing methods for editing 2D curve networks do not satisfy these criteria. We propose a targeted As-Locally-Uniform-as-Possible (ALUP) vector clip-art reshaping method that satisfies the properties above. We formulate the computation of the desired output network as the solution of a constrained variational optimization problem. We effectively compute the desired solution by casting this continuous problem as a minimization of a non-linear discrete energy function, and obtain the desired minimizer by using a custom iterative solver. We validate our method via perceptual studies comparing our results to those created via algorithmic alternatives and manually generated ones. Participants preferred our results over the closest alternative by a ratio of 6 to 1.

Fast MATLAB evaluation of nonlinear energies using FEM in 2D and 3D: Nodal elements

Nonlinear energy functionals appearing in the calculus of variations can be discretized by the finite element (FE) method and formulated as a sum of energy contributions from local elements. A fast evaluation of energy functionals containing the first order gradient terms is a central part of this contribution. We describe a vectorized implementation using the simplest linear nodal (P1) elements in which all energy contributions are evaluated all at once without the loop over triangular or tetrahedral elements. Furthermore, in connection to the first-order optimization methods, the discrete gradient of energy functional is assembled in a way that the gradient components are evaluated over all degrees of freedom all at once. The key ingredient is the vectorization of exact or approximate energy gradients over nodal patches. It leads to a time-efficient implementation at higher memory-cost. Provided codes in MATLAB related to 2D/3D hyperelasticity and 2D p-Laplacian problem are available for download and structured in a way it can be easily extended to other types of vector or scalar forms of energies.

Geometric linearization of theories for incompressible elastic materials and applications

We derive geometrically linearized theories for incompressible materials from nonlinear elasticity theory in the small displacement regime. Our nonlinear stored energy densities may vary on the same (small) length scale as the typical displacements. This allows for applications to multiwell energies as, e.g. encountered in martensitic phases of shape memory alloys and models for nematic elastomers. Under natural assumptions on the asymptotic behavior of such densities we prove Gamma-convergence of the properly rescaled nonlinear energy functionals to the relaxation of an effective model. The resulting limiting theory is geometrically linearized in the sense that it acts on infinitesimal displacements rather than finite deformations, but will in general still have a limiting stored energy density that depends in a nonlinear way on the infinitesimal strains. Our results, in particular, establish a rigorous link of existing finite and infinitesimal theories for incompressible nematic elastomers.

Automatic Sitting Pose Generation for Ergonomic Ratings of Chairs.

Human poses play a critical role in human-centric product design. Despite considerable researches on pose synthesis and pose-driven product design, most of them adopt the simple stick figure model that captures only skeletons rather than real body geometries and do not link human poses to the environment (e.g., chairs for sitting). This paper focuses on user-tailored ergonomic design and rating of chairs using scanned human geometries. Fully utilizing the anthropometric information of the human models, our method considers more ergonomic guidelines of chair design (such as pressure distribution and support intensity) and links the geometry of 3D chair models and human-to-chair interactions into the pose deformation constraints of the human avatars. The core of our method is a pose generation algorithm which rigs the user's successive poses through coarse- and fine-level pose deformations. We define a non-linear energy function with contact, collision, and joint limit terms, and solve it using a hill-climbing algorithm. The fitting results allow us to quantitatively evaluate the chair model in terms of various ergonomic criteria. Our method is flexible and effective and can be applied to users with varying body shapes and a wide range of chairs. Moreover, the proposed technique can be easily extended to other furniture, such as desk, bed, and cabinet. Extensive evaluations and a user study demonstrate the efficiency and advantages of the proposed virtual fitting method. Given that our method avoids tedious on-site trying, facilitates the exploration/evaluation of various chair products, and provides valuable feedback for the designers and manufacturers to deliver customized products, it is ideal for online shopping of chairs.

Drone routing with energy function: Formulation and exact algorithm

Drone delivery is known as a potential contributor in improving efficiency and alleviating last-mile delivery problems. For this reason, drone routing and scheduling has become a highly active area of research in recent years. Unlike the vehicle routing problem, however, designing drones’ routes is challenging due to multiple operational characteristics including multi-trip operations, recharge planning, and energy consumption calculation. To fill some important gaps in the literature, this paper solves a multi-trip drone routing problem, where drones’ energy consumption is modeled as a nonlinear function of payload and travel distance. We propose adding logical cuts and subgradient cuts in the solution process to tackle the more complex nonlinear (convex) energy function, instead of using the linear approximation method as in the literature, which can fail to detect infeasible routes due to excess energy consumption. We use a 2-index formulation to model the problem and develop a branch-and-cut algorithm for the formulation. Benchmark instances are first generated for this problem. Numerical tests indicate that even though the original model is nonlinear, the proposed approach can solve large problems to optimality. In addition, in multiple instances, the linear approximation model yields routes that under the nonlinear energy model would be energy infeasible. Use of a linear approximation for drone energy leads to differences in energy consumption of about 9% on average compared to the nonlinear energy model.

Sparse Graph Regularized Mesh Color Edit Propagation.

Mesh color edit propagation aims to propagate the color from a few color strokes to the whole mesh, which is useful for mesh colorization, color enhancement and color editing, etc. Compared with image edit propagation, luminance information is not available for 3D mesh data, so the color edit propagation is more difficult on 3D meshes than images, with far less research carried out. This paper proposes a novel solution based on sparse graph regularization. Firstly, a few color strokes are interactively drawn by the user, and then the color will be propagated to the whole mesh by minimizing a sparse graph regularized nonlinear energy function. The proposed method effectively measures geometric similarity over shapes by using a set of complementary multiscale feature descriptors, and effectively controls color bleeding via a sparse ℓ1 optimization rather than quadratic minimization used in existing work. The proposed framework can be applied for the task of interactive mesh colorization, mesh color enhancement and mesh color editing. Extensive qualitative and quantitative experiments show that the proposed method outperforms the state-of-the-art methods.

Spatial Memory and Taxis-Driven Pattern Formation in Model Ecosystems

Mathematical models of spatial population dynamics typically focus on the interplay between dispersal events and birth/death processes. However, for many animal communities, significant arrangement in space can occur on shorter timescales, where births and deaths are negligible. This phenomenon is particularly prevalent in populations of larger, vertebrate animals who often reproduce only once per year or less. To understand spatial arrangements of animal communities on such timescales, we use a class of diffusion–taxis equations for modelling inter-population movement responses between N ge 2 populations. These systems of equations incorporate the effect on animal movement of both the current presence of other populations and the memory of past presence encoded either in the environment or in the minds of animals. We give general criteria for the spontaneous formation of both stationary and oscillatory patterns, via linear pattern formation analysis. For N=2, we classify completely the pattern formation properties using a combination of linear analysis and nonlinear energy functionals. In this case, the only patterns that can occur asymptotically in time are stationary. However, for N ge 3, oscillatory patterns can occur asymptotically, giving rise to a sequence of period-doubling bifurcations leading to patterns with no obvious regularity, a hallmark of chaos. Our study highlights the importance of understanding between-population animal movement for understanding spatial species distributions, something that is typically ignored in species distribution modelling, and so develops a new paradigm for spatial population dynamics.

Stabilization of infinite dimensional port-Hamiltonian systems by nonlinear dynamic boundary control

The conditions for existence of solutions and stability, asymptotic and exponential, of a large class of boundary controlled systems on a 1D spatial domain subject to nonlinear dynamic boundary actuation are given. The consideration of such class of control systems is motivated by the use of actuators and sensors with nonlinear behavior in many engineering applications. These nonlinearities are usually associated to large deformations or the use of smart materials such as piezo actuators and memory shape alloys. Including them in the controller model results in passive dynamic controllers with nonlinear potential energy function and/or nonlinear damping forces. First it is shown that under very natural assumptions the solutions of the partial differential equation with the nonlinear dynamic boundary conditions exist globally. Secondly, when energy dissipation is present in the controller, then it globally asymptotically stabilizes the partial differential equation. Finally, it is shown that assuming some additional conditions on the interconnection and on the passivity properties of the controller (consistent with physical applications) global exponential stability of the closed-loop system is achieved.

Robust mesh deformation with salient features preservation

Triangular meshes often contain a few salient features. Traditional deformation algorithms mainly preserve the local details and volume, thus producing unnatural results. This paper proposes a robust and effective algorithm to prevent the distortion of salient features. Firstly, the salient features can be automatically extracted through saliency-based clustering and aggregation. A nonlinear energy function is then minimized to make the salient features behave rigidly to retain the shape. Finally, for the robustness of the minimization, we generate a coarse solid subspace around the input mesh, and carry out the energy minimization in this subspace. Experiments show that our algorithm can preserve the salient features and obtain visual-pleasing results.

A Multi-Constrained Routing Algorithm for Software Defined Network Based on Nonlinear Annealing

Multi-constrained Quality of Service (QoS) routing algorithm is always a difficult problem in routing research area, which is a NP problem. Software Defined Network (SDN) is a new network architecture, in which there’s few research on QoS routing. This paper generates the whole network virtual topology according to the characteristics of SDN, and based on the principle of simulated annealing, proposes a nonlinear annealing algorithm, which adapts to the SDN. Firstly we simplify the network topology by Dijkstra-like algorithm, and then introduce the nonlinear energy function, and then iterate the initial solution according to the simulated temperature, until find a feasible path from the source node to the destination node which satisfies the condition. Experimental results show that this algorithm has a higher success rate, better expansibility of network size, and better transplantation for the SDN than traditional QoS routing algorithm

A meta-cognitive interval type-2 fuzzy inference system and its projection based learning algorithm

A meta-cognitive interval type-2 neuro-fuzzy inference system (McIT2FIS) based classifier and its projection based learning algorithm is presented in this paper. McIT2FIS consists of two components, namely, a cognitive component and a meta-cognitive component. The cognitive component is an interval type-2 neuro-fuzzy inference system (IT2FIS) represented as a six layered adaptive network realizing Takagi-Sugeno-Kang type inference mechanism. A self-regulatory learning mechanism forms the meta-cognitive component. IT2FIS begins with zero rules, and rules are added and updated depending on the prediction error and relative knowledge contained the current sample. As each sample is presented to the network, the meta-cognitive component monitors the hinge-loss error and class-specific spherical potential of the current sample to decide what-to-learn, when-to-learn and how-to-learn them, efficiently. When a new rule is added or when an existing rule is updated, a projection based learning algorithm computes the optimal output weights with least computational effort by finding analytical minima of the nonlinear energy function. It uses class specific criterion and sample overlap criterion to estimate the network parameters corresponding to the minimum energy point of the error function. Moreover, consistently under - performing rules are pruned from the network leading to a compact network. The performance of McIT2FIS is first evaluated on a set of benchmark classification problems from UCI machine learning repository. A tenfold cross validation based performance comparison with other state-of-the-art approaches indicates its improved performance. Next, its performance is evaluated on detection of attention deficiency hyperactivity disorder (ADHD) in children. The aim of this study is to classify a child as having typically developing controls or as an ADHD patient. Voxel based features extracted from amygdala region of the brain is employed in this study. The network is trained and tested on samples obtained from ADHD-200 consortium dataset consisting of 941 subjects. The performance comparison with standard support vector machine shows that McIT2FIS has superior classification ability than SVM in diagnosing ADHD.

Sobolev-like cones of trace-class operators on unbounded domains: Interpolation inequalities and compactness properties

In this paper we extend the compactness properties for trace-class operators obtained by Dolbeault, Felmer and Mayorga-Zambrano to a smooth unbounded domain Ω⊆Rd, d≥3. We consider V, a non-negative potential on Ω that blows up at infinity, and the normed space HV(Ω)={u∈H01(Ω):‖u‖V2=∫Ω(∣∇u(x)∣2+∣u(x)∣2V(x))dx<∞}. A positive self-adjoint trace-class operator R belongs to the Sobolev-like cone HV,+1 if (ψi,R)N⊆HV(Ω) and 《R》V=∑i=1∞νi,R‖ψi,R‖V2<∞, where (νi,R)i∈N is the sequence of occupation numbers of R and (ψi,R)i∈N⊆L2(Ω) is a corresponding Hilbertian basis of eigenfunctions. We prove that a sequence in HV,+1, bounded in energy 《⋅》V, has a subsequence that converges in trace norm; this is analogous to the classical Sobolev immersion H1(Ω)⊆L2(Ω). We prove the existence of lower bounds for nonlinear free energy functionals and, by doing so, we establish Lieb–Thirring type inequalities as well as some Gagliardo–Nirenberg type interpolation inequalities; then our compactness result is applied to minimize nonlinear free energy functionals working on HV,+1.

Meta-cognitive RBF Network and its Projection Based Learning algorithm for classification problems

‘Meta-cognitive Radial Basis Function Network’ (McRBFN) and its ‘Projection Based Learning’ (PBL) algorithm for classification problems in sequential framework is proposed in this paper and is referred to as PBL-McRBFN. McRBFN is inspired by human meta-cognitive learning principles. McRBFN has two components, namely the cognitive component and the meta-cognitive component. The cognitive component is a single hidden layer radial basis function network with evolving architecture. In the cognitive component, the PBL algorithm computes the optimal output weights with least computational effort by finding analytical minima of the nonlinear energy function. The meta-cognitive component controls the learning process in the cognitive component by choosing the best learning strategy for the current sample and adapts the learning strategies by implementing self-regulation. In addition, sample overlapping conditions are considered for proper initialization of new hidden neurons, thus minimizes the misclassification. The interaction of cognitive component and meta-cognitive component address the what-to-learn, when-to-learn and how-to-learn human learning principles efficiently. The performance of the PBL-McRBFN is evaluated using a set of benchmark classification problems from UCI machine learning repository and two practical problems, viz., the acoustic emission signal classification and the mammogram for cancer classification. The statistical performance evaluation on these problems has proven the superior performance of PBL-McRBFN classifier over results reported in the literature.

Linear Γ-limits of multiwell energies in nonlinear elasticity theory

We derive linearized theories from nonlinear elasticity theory for multiwell energies. Under natural assumptions on the nonlinear stored energy densities, the properly rescaled nonlinear energy functionals are shown to Γ-converge to the relaxation of a corresponding linearized model. Minimizing sequences of problems with displacement boundary conditions and body forces are investigated and found to correspond to minimizing sequences of the linearized problems. As applications of our results we discuss the validity and failure of a formula that is widely used to model multiwell energies in the regime of linear elasticity. Applying our convergence results to the special case of single well densities, we also obtain a new strong convergence result for the sequence of minimizers of the nonlinear problem.

Relativistic random-phase approximation in axial symmetry

Covariant density functional theory, in the framework of self-consistent relativistic mean field (RMF) and relativistic random-phase approximation (RRPA), is for the first time applied to axially deformed nuclei. The fully self-consistent RMF+RRPA equations are posed for the case of axial symmetry and nonlinear energy functionals and solved with the help of a new parallel code. Formal properties of RPA theory are studied and special care is taken to validate the proper decoupling of spurious modes and their influence on the physical response. Sample applications to the magnetic and electric dipole transitions in $^{20}\mathrm{Ne}$ are presented and analyzed.

2D shape deformation using nonlinear least squares optimization

This paper presents a novel 2D shape deformation algorithm based on nonlinear least squares optimization. The algorithm aims to preserve two local shape properties: the Laplacian coordinates of the boundary curve and the local area of the shape interior, which are together represented in a non-quadratic energy function. An iterative Gauss–Newton method is used to minimize this nonlinear energy function. The result is an interactive shape deformation system that can achieve physically plausible results that are difficult to achieve with previous linear least squares methods. In addition to this algorithm that preserves local shape properties, we also introduce a scheme to preserve the global area of the shape, which is useful for deforming incompressible objects.

A massively parallel approach to deformable matching of 3D medical images via stochastic differential equations

The deformable matching of 3D medical images remains a difficult problem due to the high dimension of both geometric transformations and data. The matching problem is usually expressed as the minimization of a highly non-linear energy (objective) function, yielding a hard, computationally intensive, optimization problem. This paper presents a comprehensive parallel approach that yields computation times compatible with clinical routine. The image matching is based on the simulation of stochastic differential equations, enabling the optimization of the global objective function, through an annealing process. The resulting algorithm allows a fully parallel sampling of the parameters to be optimized. Due to the large number of parameters involved in deformable matching, this approach is naturally suited to massively parallel implementations. We present implementation issues and timing analysis on an MIMD parallel processing computer (SGI Origin 2000). The performances of the approach are assessed on real data, using 3D brain MR images from different individuals. Beside yielding accurate registrations, the parallel algorithm exhibits excellent relative speedups.