Mixed Formulation Research Articles

Multiclass optimal classification trees with SVM-splits

In this paper we present a novel mathematical optimization-based methodology to construct tree-shaped classification rules for multiclass instances. Our approach consists of building Classification Trees in which, except for the leaf nodes, the labels are temporarily left out and grouped into two classes by means of a SVM separating hyperplane. We provide a Mixed Integer Non Linear Programming formulation for the problem and report the results of an extended battery of computational experiments to assess the performance of our proposal with respect to other benchmarking classification methods.

Linear or mixed integer programming in long-term energy systems modeling – A comparative analysis for a local expanding heating system

Most computer models used in energy systems optimization modeling studies are formulated using linear equations. However, since linear formulations do not always well reflect real-world conditions, they may not always be adequate as policy and support tools. This is particularly the case for local system studies attempting to represent technologies at the individual scale, as in the case for local heating system modeling. Thus, the aim of this paper is to investigate differences in the resulting heating solutions and model solution times for a local expanding heating system. Three different investment cost structures for individual and district heating solutions for the heating of new housing are investigated using linear and mixed integer linear programming. The results show that the use of district heating is higher for the cost structures that use mixed integer linear programming than it is for the linear cost structures. This result is attributed mainly to the fact that individual air-to-water heat pumps benefit from the linear equation formulation due to its high coefficient of performance during summertime. This finding is important to consider when modeling local energy systems. The solution time is, however, significantly shorter for the linear formulations than for the mixed integer linear formulations.

A selectively reduced degree basis for efficient mixed nonlinear isogeometric beam formulations with extensible directors

The effect of higher order continuity in the solution field by using NURBS basis function in isogeometric analysis (IGA) is investigated for an efficient mixed finite element formulation for elastostatic beams. It is based on the Hu–Washizu variational principle considering geometrical and material nonlinearities. Here we present a reduced degree of basis functions for the additional fields of the stress resultants and strains of the beam, which are allowed to be discontinuous across elements. This approach turns out to significantly improve the computational efficiency and the accuracy of the results. We consider a beam formulation with extensible directors, where cross-sectional strains are enriched to avoid Poisson locking by an enhanced assumed strain method. In numerical examples, we show the superior per degree-of-freedom accuracy of IGA over conventional finite element analysis, due to the higher order continuity in the displacement field. We further verify the efficient rotational coupling between beams, as well as the path-independence of the results.

Randomized neural network with Petrov–Galerkin methods for solving linear and nonlinear partial differential equations

We present a new approach for solving partial differential equations (PDEs) based on randomized neural networks and the Petrov–Galerkin method, which we call the RNN-PG methods. This method uses randomized neural networks to approximate unknown functions and allows for a flexible choice of test functions, such as finite element basis functions, Legendre or Chebyshev polynomials, or neural networks. We apply the RNN-PG methods to various problems, including Poisson problems with primal or mixed formulations, and time-dependent problems with a space–time approach. This paper is adapted from the work originally posted on arXiv.com by the same authors (arXiv:2201.12995, Jan 31, 2022). The new ingredients include non-linear PDEs such as Burger’s equation and a numerical example of a high-dimensional heat equation. Numerical experiments show that the RNN-PG methods can achieve high accuracy with a small number of degrees of freedom. Moreover, RNN-PG has several advantages, such as being mesh-free, handling different boundary conditions easily, solving time-dependent problems efficiently, and solving high-dimensional problems quickly. These results demonstrate the great potential of the RNN-PG methods in the field of numerical methods for PDEs.

Interpolation operators for parabolic problems

We introduce interpolation operators with approximation and stability properties suited for parabolic problems in primal and mixed formulations. We derive localized error estimates for tensor product meshes (occurring in classical time-marching schemes) as well as locally in space-time refined meshes.

A mixed finite element method for nonlinear radiation–conduction equations in optically thick anisotropic media

We propose a new mixed finite element formulation for solving radiation–conduction heat transfer in optically thick anisotropic media. At this optical regime, the integro-differential equations for radiative transfer can be replaced by the simplified PN approximations using an asymptotic analysis. The conductivity is assumed to be nonlinear depending on the temperature along with anisotropic absorption and scattering depending on both the direction and location variables. The simplified PN approximations are enhanced by considering a diffusion tensor capable of describing anisotropic radiative heat transfer. In the present study, we investigate the performance of the unified and mixed formulations combining cubic P3, quadratic P2, and linear P1 finite elements to approximate the temperature in the simplified P3 model. To demonstrate the performance of the proposed methodology, three-dimensional examples of nonlinear radiation–conduction equations in optically thick anisotropic media are presented. The obtained numerical results demonstrate the accuracy and efficiency of the proposed mixed finite element formulation over the conventional unified finite element formulation to accurately solve the simplified P3 equations in anisotropic media.

Advanced discretization techniques for hyperelastic physics-augmented neural networks

In the present work, advanced spatial and temporal discretization techniques are tailored to hyperelastic physics-augmented neural networks, i.e., neural network based constitutive models which fulfill all relevant mechanical conditions of hyperelasticity by construction. The framework takes into account the structure of neural network-based constitutive models, in particular, that their derivatives are more complex compared to analytical models. The proposed framework allows for convenient mixed Hu–Washizu like finite element formulations applicable to nearly incompressible material behavior. The key feature of this work is a tailored energy–momentum scheme for time discretization, which allows for energy and momentum preserving dynamical simulations. Both the mixed formulation and the energy–momentum discretization are applied in finite element analysis. For this, a hyperelastic physics-augmented neural network model is calibrated to data generated with an analytical potential. In all finite element simulations, the proposed discretization techniques show excellent performance. All of this demonstrates that, from a formal point of view, neural networks are essentially mathematical functions. As such, they can be applied in numerical methods as straightforwardly as analytical constitutive models. Nevertheless, their special structure suggests to tailor advanced discretization methods, to arrive at compact mathematical formulations and convenient implementations.

Potential Use of Wheat Straw, Grape Pomace, Olive Mill Wastewater and Cheese Whey in Mixed Formulations for Silage Production

Two experiments were conducted to investigate the chemical and fermentative characteristics of by-product-mixed silages consisting of wheat straw (WS), grape pomace (GP), olive mill wastewater (OMWW) and cheese whey (CW) at 7, 30 and 90 days. The silage formulations were based on a ratio of 60% solids (WS + GP) and 40% liquids (CW + OMWW), with the addition of water (W) where necessary to achieve 40% of liquids. In experiment 1, the effects of the inclusion of GP or CW in a mixture of WS and OMWW were studied according to two silage formulations: SIL-A, WS40% + OMWW5% + GP20% + W35%; SIL-B, WS60% + OMWW5% + CW35%. In experiment 2, the effects of two levels of CW and the inclusion of OMWW in mixed silages based on WS, GP, and CW were studied according to four silage formulations: SIL-C, WS40% + GP20% + CW20% + W20%; SIL-D, WS40% + GP20% + CW20% + OMWW5% + W15%; SIL-E, WS40% + GP20% + CW35% + W5%; SIL-F, WS40% + GP20% + CW35% + OMWW5%. In experiment 1, the silage formulation affected the chemical composition showing a greater (p < 0.05) content of DM in SIL-B; crude protein, ether extract and ADL contents were higher (p < 0.05) in SIL-A. In experiment 2, no differences (p > 0.05) in the chemical characteristics of the silages were found. In both of the experiments, the chemical composition and total phenol content did not change (p > 0.05) during the ensiling period. Fermentative characteristics were not affected (p > 0.05) by the by-product combination nor the ensiling period and proved to be adequate for good-quality silages. The Flieg’s scores at D30 and D90 were greater than a 100 score in all the experimental silages, leading to the conclusion that WS, GP, OMWW and CW can be effective for producing silage.

An integer programming approach for the hyper-rectangular clustering problem with axis-parallel clusters and outliers

We present a mixed integer programming formulation for the problem of clustering a set of points in Rd with axis-parallel clusters, while allowing to discard a pre-specified number of points, thus declared to be outliers. We identify a family of valid inequalities separable in polynomial time, we prove that some inequalities from this family induce facets of the associated polytope, and we show that the dynamic addition of cuts coming from this family is effective in practice.

Fourier analysis of membrane locking and unlocking

Despite four decades of research membrane locking remains an important issue hindering the development of effective beam and shell finite elements. In this article, we utilize Fourier analysis of the complete spectrum of natural vibrations and propose a criterion to identify and evaluate the severity of membrane locking. To demonstrate our approach, we utilize primal and mixed Galerkin formulations applied to a circular Euler–Bernoulli ring discretized using uniform, periodic B-splines. By analytically computing the discrete Fourier operators, we obtain an exact representation of the normalized error across the entire spectrum of eigenvalues. Our investigation addresses key questions related to membrane locking, including mode susceptibility, the influence of polynomial order, and the impact of shell/beam thickness and radius of curvature. Furthermore, we compare the effectiveness of mixed and primal Galerkin methods in mitigating locking. By providing insights into the parameters affecting locking and introducing a criterion to evaluate its severity, this research contributes to the development of improved numerical methods for thin beams and shells.

Hydrodynamics simulation of red blood cells: Employing a penalty method with double jump composition of lower order time integrator

We propose a numerical framework tailored for simulating the dynamics of vesicles with inextensible membranes, which mimic red blood cells, immersed in a non‐Newtonian fluid environment. A penalty method is proposed to handle the inextensibility constraint by relaxation, allowing a simple computer implementation and an affordable computational load compared to the full mixed formulation. To handle the high‐order derivatives in the stress jump across the membrane, which arise due to the high geometric order of the Helfrich functional, we employ higher degree finite elements for spatial discretization. The time integration scheme relies on the double composition of the Crank–Nicolson scheme to achieve faster fourth‐order convergence behavior. Additionally, an adaptive time‐stepping strategy based on a third‐order temporal integration error estimation is implemented. We address the main features of the proposed method, which is benchmarked against existing numerical and experimental results. Furthermore, we investigate the influence of non‐Newtonian rheology on the system dynamics.

A variationally consistent contact formulation based on a mixed interpolation point method and isogeometric discretization

This work enhances the robustness of the standard displacement-based (penalty Gauss-point-to-segment) contact formulation by applying the so-called Mixed Interpolation Point method (MIP). We consider a smooth isogeometric discretization of the contact surface in order to avoid various artificial discontinuities of the normal contact gap function. Like any existing MIP-enhanced formulation in terms of implementation, the proposed MIP contact formulation only requires a minor modification from an existing implementation of the standard contact formulation. Nevertheless, the improvement in robustness is shown to be quite significant, which often enables the simulation of contact problems with a larger load step size for efficiency and/or a larger penalty parameter for accuracy in the contact constraint. The formalism of the proposed MIP contact method is based on the idea of relaxing the contact constitution at integration points. To this end, at first the contact pressure is considered as an additional unknown apart from the displacement field, and the perturbed Lagrange multiplier potential is used to enforce the contact constraint. The MIP method then eliminates the contact pressure unknown directly at integration points, instead of discretizing it as it is done in the standard mixed contact formulations. As a result, the residual vector is identical to the standard displacement-based contact formulation. However, the resulting tangent stiffness matrix is different, as the MIP tangent is now based on an extrapolation of the contact pressure iteratively over Newton iterations. Several challenging numerical examples are presented to illustrate the accuracy, robustness and efficiency of the proposed formulation.

Rebalancing in bike sharing systems: Application to the Lisbon case study

This work presents a mathematical programming formulation for the static Bike Rebalancing Problem to meet the practical constraints faced by the operators of Bike Sharing Systems. Adaptating a Mixed Integer Linear Programming formulation, it now includes different types of vehicles and bicycles, the autonomy of each vehicle, the duration of its route and the stations that require inspection actions. An intermediate station is introduced to temporarily store bicycles, relaxing the restrictions imposed by the vehicles’ capacity. After incorporating different acceleration procedures, a Branch-and-Cut scheme is run for 30 min to find optimized routes for the reposition vehicle on five real instances of the Lisbon Bike Sharing System. The computed routes resulted in a 30% to 75% decrease in total length, when compared to the real routes. This decrease is mainly attributed to the integration of all the mentioned factors in one global model, encouraging its future implementation.

Nonlinear analysis of reinforced concrete slabs using a quasi-3D mixed finite element formulation

This study presents an efficient quasi-3D mixed finite element (MFE) formulation for the materially nonlinear analysis of reinforced concrete (RC) slabs based on a refined layered global–local plate theory. The cross-section of the RC slab is divided into a series of concrete and steel layers. Each layer is treated as a plate structure and mechanical properties of the steel and concrete materials are individually applied to each layer. In addition to unknown variables of the displacement field, out-of-plane stress components are also assumed as independent field variables in the presented MFE formulation. It makes it possible to obtain the out-of-plane shear stresses at each node directly from constitutive equations with the enough accuracy. A four‐node rectangular element ensuring the C1‐continuity is used for discretizing the domain of the RC slab. The governing equations are derived based on a partially mixed-field variational principle. The accuracy and efficiency of the presented MFE have been investigated through some benchmark examples. Numerical tests show that the presented MFE yields results with sufficient accuracy at a low computational cost.

Simulation of hydrogen embrittlement of steel using mixed nonlocal finite elements

A new simulation strategy to model hydrogen embrittlement based on a multi-field finite element using displacements, pressure, volume variation, nonlocal damage variables and lattice hydrogen concentration as unknowns is proposed. The material is described using a modified GTN model, which includes the description of hydrogen enhanced decohesion (HEDE). The finite element problem is solved using a fully implicit formulation. Numerical problems related to volumetric locking are solved using a mixed pressure/volume variation formulation. The use of the mixed formulation allows a straightforward evaluation of the pressure gradient, which drives hydrogen diffusion. Mesh size dependence is solved using an implicit gradient nonlocal formulation. Two variables are used to represent damage: the plastic volume variation and the accumulated plastic strain, controlling nucleation. The model is used to simulate an existing experimental database (Moro et al., 2010; Briottet et al., 2012) including tensile, fracture toughness and pressurized disk tests. The model, after adjusting the various coefficients, can represent the main experimental findings: the effect of deformation rate on the failure of tensile specimens, the transition from surface to internal fracture with increasing deformation rate, the sharp toughness drop under hydrogen (CT specimen), the fracture location and the effect of the pressurization rate in the case of the tests on disks.

RP-HPLC Method Development and Validation for Estimation of Bisoprolol and Hydrochlorothiazide

The present study was to develop a rapid and sensitive method for the analysis of Bisoprolol and Hydrochlorothiazide by different analytical methods such as UV spectrophotometry and RP-HPLC in pharmaceutical dosage forms by using the most commonly employed C-18 column with UV-detection and validate it as per ICH guidelines. The retention time for Bisoprolol and Hydrochlorothiazide was found to be 2.1 min and 4.01 min respectively. A Resolution of greater than 2 was observed. The low % RSD values (≤ 2) indicated that the method was precise and accurate. The mean recoveries were found in the range of 99.1 – 99.9% w/w. Our approach was specific with good accuracy and precision when evaluating the commercial formulations without the interference of excipients and other additives. This method can be used to calculate the amounts of Bisoprolol and Hydrochlorothiazide in both single and mixed pharmaceutical formulations, as well as bulk samples.

Tensor product P-splines using a sparse mixed model formulation

A new approach to represent P-splines as a mixed model is presented. The corresponding matrices are sparse allowing the new approach can find the optimal values of the penalty parameters in a computationally efficient manner. Whereas the new mixed model P-splines formulation is similar to the original P-splines, a key difference is that the fixed effects are modelled explicitly, and extra constraints are added to the random part of the model. An important feature ensuring that the entire computation is fast is a sparse implementation of the Automated Differentiation of the Cholesky algorithm. It is shown by means of two examples that the new approach is fast compared to existing methods. The methodology has been implemented in the R-package LMMsolver available on CRAN ( https://CRAN.R-project.org/package=LMMsolver ).

Strong cutting planes for the capacitated multi-pickup and delivery problem with time windows

In this paper, we propose an improved 2-index mixed integer linear programming formulation for capacitated multi-pickup and delivery problems with time windows. We develop new families of valid inequalities, present some simple separation heuristics and solve the benchmark problems using a cutting plane approach. Specifically, we propose and demonstrate the effectiveness of incompatible request sets based valid inequalities, exploiting thereby the time window and capacity constraints. Further, we introduce two sub-families of these inequalities: node-based and request-based incompatible request set cuts, along with their separation heuristics. The computational results show that the incompatible request set cuts are highly useful as cutting planes for the benchmark instances. Overall, there is a marked improvement in solution performance vis-à-vis the different formulations and solution approaches used in extant literature. The approach in this paper is able to solve 12%, 37% and 43% more instances to optimality, improve the root gap by 18%, 33% and 36%, and final gap by 36%, 45% and 43% vis-à-vis the asymmetric representative formulation, three-index and the two-index formulations respectively.

Encoded information of mixed correlations: the views from one dimension higher

After reviewing the JT gravity, we discuss the four saddles in the mixed correlation measures of black holes Hawking radiation in the setup of geometric evaporation of [1]. By looking from 1d higher point of view and partial dimensional reduction, we examine the phase structures and the universalities for these four saddles. We also discuss the behavior of quantum error correction codes for each of these four phases, reaching to consistent results. Then, instead of dimension reduction between Einstein gravity and JT, we try to explore the connections between partition functions and saddles of 3d Chern-Simons and 2d BF theories, 2d Liouville and 2d Wess-Zumino-Witten models, and also the dimensionally reduced 1d Schwarzian and 1d particles on group. We specifically sketch on the connections between these theories in the setup of mixed correlations and island formulation.

The cumulative school bus routing problem: Polynomial‐size formulations

AbstractThis article introduces the cumulative school bus routing problem, which concerns the transport of students from a school using a fleet of identical buses. The objective of the problem is to select a drop‐off point for each student among potential locations within a certain walking distance and to generate routes such that the sum of arrival times of all students from their school to their homes is minimized. The article describes six polynomial‐size mixed integer linear programming formulations based on original and auxiliary graphs, and the formulations are numerically compared on real instances. The article reports the results of computational experiments performed to evaluate the performance of the proposed models.