Arbitrary Constraints Research Articles

Construction of Uniform Designs over a Domain with Linear Constraints

Uniform design is a powerful and robust experimental methodology that is particularly advantageous for multidimensional numerical integration and high-level experiments. As its applications expand across diverse disciplines, the theoretical foundation of uniform design continues to evolve. In real-world scenarios, experimental factors are often subject to one or more linear constraints, which pose challenges in constructing efficient designs within constrained high-dimensional experimental spaces. These challenges typically require sophisticated algorithms, which may compromise uniformity and robustness. Addressing these constraints is critical for reducing costs, improving model accuracy, and identifying global optima in optimization problems. However, existing research primarily focuses on unconstrained or minimally constrained hypercubes, leaving a gap in constructing designs tailored to arbitrary linear constraints. This study bridges this gap by extending the inverse Rosenblatt transformation framework to develop innovative methods for constructing uniform designs over arbitrary hyperplanes and hyperspheres within unit hypercubes. Explicit construction formulas for these constrained domains are derived, offering simplified calculations for practitioners and providing a practical solution applicable to a wide range of experimental scenarios. Numerical simulations demonstrate the feasibility and effectiveness of these methods, setting a new benchmark for uniform design in constrained experimental regions.

Composable Constraint Models for Permutation Enumeration

Constraint programming (CP) is a powerful tool for modeling mathematical concepts and objects and finding both solutions or counter examples. One of the major strengths of CP is that problems can easily be combined or expanded. In this paper, we illustrate that this versatility makes CP an ideal tool for exploring problems in permutation patterns. We declaratively define permutation properties, permutation pattern avoidance and containment constraints using CP and show how this allows us to solve a wide range of problems. We show how this approach enables the arbitrary composition of these conditions, and also allows the easy addition of extra conditions. We demonstrate the effectiveness of our techniques by modelling the containment and avoidance of six permutation patterns, eight permutation properties and measuring five statistics on the resulting permutations. In addition to calculating properties and statistics for the generated permutations, we show that arbitrary additional constraints can also be easily and efficiently added. This approach enables mathematicians to investigate permutation pattern problems in a quick and efficient manner. We demonstrate the utility of constraint programming for permutation patterns by showing how we can easily and efficiently extend the known permutation counts for a conjecture involving the class of $1324$ avoiding permutations. For this problem, we expand the enumeration of $1324$-avoiding permutations with a fixed number of inversions to permutations of length 16 and show for the first time that in the enumeration there is a pattern occurring which follows a unique sequence on the Online Encyclopedia of Integer Sequences.

Scarcity Poetics: Christian Bök’s Eunoia and the Economics of Literary Value

Christian Bök’s Canadian bestseller Eunoia (2001) is an ideal study in how the romantic notion of literary value actually abides economic theories. Eunoia is a collection of five prose-poems each written using only one vowel grapheme (A, E, I, O, or U). These arbitrary material production constraints work just like economic sanctions, artificially inflating the scarcity—and hence value—of the text, both commercially and literarily. Discourse analysis of surrounding debates reveals that detractions and praises alike abide the same basic supply-and-demand logic: e.g., economic theories of (relative) marginal utility, as applied by Lee Erickson in The Economy of Literary Forms (1996). Building on Erickson’s thesis of how publication media costs shaped literary form and content, on Mary Poovey’s history of literary value’s origin in economic value, and on other efforts to combine literary study with economics, this essay applies economic theory to a rare opportunity to generalize the relation of literary material, form, and content. Conclusively, scarcity poetics/tactics are not just unique to Eunoia, but fundamental to all literary form. Eunoia’s extreme manipulation of linguistic materials isolates the general mechanism by which literary value is produced.

Analytical solutions for large-deflection bending of variable-thickness inhomogeneous functional graded composite circular plates with parameterized boundaries under hygro-thermo-mechanical loads

A brand-new hygro-thermo-mechanical bending model of the inhomogeneous concave and convex composite circular plates having varying thickness undergoing large deformation resting on nonlinear tri-parameter spring and shear elastic layers is proposed. Three-dimensional analytical hygrothermal field of circular plate is demonstrated with non-uniform thermal and moisture diffusion coefficients in axial variation, while thickness and elastic module of material are modeled in quadratically change and sufficiently summarized in normalization. The circled boundaries with arbitrary translational and rotational constraints are parameterized with feasible region of elastic parameters highlighted in rectangular domain by Linear Programming method. Two reduced-order integro-differential governing equations for the composite circular plates under extreme load have been derived, while analytical bending solutions are obtained by an employed homotopy-based analytical scheme with accuracy verified and convergence accelerated by truncation and iteration. Whether thickness or elastic module of composite material is variable, the final outcome on plate structural strength is the discrepancy of bending stiffness with different load capacity, with the former revealing more sensitive than the latter under the condition of same values. Once amplitude and distribution of hygrothermal load are determined, influence on large-deflection bending of circular plates exists at a limited range of nonlinearity, while non-uniform distribution of thermal expansion and moisture concentration aggravates bending effects only within this range.

Nonlinear thermal flutter of variable angle tow curved panels with elastically restrained edges in supersonic airflow by generalized Galerkin method

Variable Angle Tows (VAT) composite materials can effectively transfer major stresses from the interior region to the edges through in-plane load redistribution, thereby significantly enhancing their structural performance. Therefore, correct handling of boundary conditions is crucial in analyzing VAT structures. Considering that most practical engineering cases involve non-ideal or non-classical boundary support, this paper proposes an efficient method to address the nonlinear thermal flutter problem of elastic supported VAT curved panels in supersonic airflow. Assuming the fiber orientation angle varies in four different ways along the x-direction, the basic equations are established based on the von Kármán large deformation plate theory and the first-order piston theory. The generalized Galerkin method, capable of handling arbitrary elastic boundary constraints, is used to solve the problem in the space domain, while the Runge-Kutta method is applied in the time domain. Numerical examples illustrate how elastically restrained conditions, fiber orientation angles, height-rise ratio, and thermal loads, along with aerodynamic pressure, influence flutter responses of VAT panels. The presented method can be used for the aeroelastic stability design of VAT curved panels under arbitrary boundary conditions.

Mathematical foundation of the master‐slave elimination for arbitrary nonlinear multi‐point constraints

AbstractNonlinear multi‐point constraints are essential in modeling various engineering problems, for example, joints undergoing large rotations or coupling of different element types in finite element analysis. They can be handled using Lagrange multipliers, the penalty method, or master‐slave elimination. The latter satisfies the constraints exactly and reduces the dimension of the resulting system of equations, which is particularly advantageous when a large number of constraints have to be considered. However, the existing schemes in literature are limited to linear constraints. Therefore, the authors introduced an extension of the method to arbitrary nonlinear constraints. A mathematically rigorous derivation of this new method is presented. The starting point is the optimization problem with constraints. It is transformed into a modified optimization problem without constraints using the implicit function theorem. For this, an appropriate selection of slave degrees of freedom (dofs) is crucial, ensuring that the Jacobian constraints meets specific conditions. This implies the consideration of three aspects: The handling of redundant constraints, the automatic selection of slave dofs, and the change of slave dofs in the context of large deformations. Additionally, it allows for the combination of the new method with existing constraint methods.

Exact solutions for anisotropic beams with arbitrary distributed loads

The general solution of elasticity for plane anisotropic beams with arbitrary constraints at ends and arbitrary normal and tangential distributed loads on surfaces is derived, which consists of internal forces (i.e., bending moment, shearing force, axial force) and their integrals and derivatives of different orders and load-independent polynomial function sequences of longitudinal coordinates. The method for determining the function sequences is established by resolving the governing equation and boundary conditions of the stress function method. For beams with an elastic symmetry plane, a method for directly determining explicit expressions for all terms of function sequences is provided. Particular solutions of examples are solved using general solution formulas, and the results align excellently with existing exact solutions. Finally, the errors in EBT and TBT when applied to beams made of different materials are analysed.

Elasticity solutions for functionally graded beams with arbitrary distributed loads

This paper derives the exact general elasticity solution for functionally graded rectangular beams subjected to arbitrary normal and tangential loads and with arbitrary end constraints. The general solution consists of bending moments and their integrals and derivatives, along with load-independent function sequences of the longitudinal coordinate. The method for determining function sequences has been established based on the stress function method. General solution formulas for stresses, strains and displacements have been derived and used to solve explicit special solutions for six cases involving concentrated forces, uniformly loads, and quadratically distributed loads with different displacement constraints scenarios. The results obtained are compared with existing exact solutions and those of Euler–Bernoulli and Timoshenko beams, and the errors of the latter two are analysed.

Leveraging physics-informed neural computing for transport simulations of nuclear fusion plasmas

For decades, plasma transport simulations in tokamaks have used the finite difference method (FDM), a relatively simple scheme to solve the transport equations, a coupled set of time-dependent partial differential equations. In this FDM approach, typically over O(105) time steps are needed for a single discharge, to mitigate numerical instabilities induced by stiff transport coefficients. It requires significant computing time as costly transport models are repeatedly called in a serial manner, proportional to the number of time steps. Additionally, the unidirectional calculations of FDM make it difficult to predict regions prior to the initial condition or apply additional temporal constraints. In this study, we discuss using a new scheme to solve plasma transport based on physics-informed neural networks (PINNs). PINN iteratively updates a function that maps spatiotemporal coordinates to plasma states, gradually reducing errors in transport equations. The required number of updates in PINNs is several orders of magnitude less than the chronological iterations in FDM. Furthermore, it is free from numerical instabilities arising from finite grids and enables more versatile semi-predictive simulations with arbitrary spatiotemporal constraints. In this paper, we discuss the features and potentials of the tokamak transport solver using PINNs through comparisons with FDM, and also its drawbacks and challenges.

Adjusting for covariates representing potential confounders, mediators, or competing predictors in the presence of measurement error: Dispelling a potential misapprehension and insights for optimal study design with nutritional epidemiology examples

Background Variables such as dietary intake are measured with error yet frequently used in observational epidemiology. Although this limitation is sometimes noted, these variables are still often modeled as covariates without formal correction or sincere dialogue about measurement unreliability potentially weakening the validity of statistical conclusions. Further, larger sample sizes increase power (bias) to detect spurious correlations. Counterintuitively, recent work suggested a non-monotonic relationship between confounder unreliability and how much controlling for the confounder reduces (or induces) bias when testing for an exposure-outcome association. If true, such non-monotonicity would be especially concerning for applications such as nutrition, where measurement reliability varies substantially, and large sample sizes are common. Methods We offer a detailed derivations of the square partial correlation between the outcome and exposure, controlling for the confounder. In our derivation, the measurement reliabilities of exposures and confounders are not arbitrarily constrained to be equal. Further, our theoretical results are investigated using simulations. Results Reassuringly, these derivations and simulations show that the counterintuitive non-monotonicity relationship between confounder unreliability and how much controlling for the confounder reduces (or induces) bias when testing for an exposure-outcome association is an artifact of the arbitrary constraint which forces the measurement reliabilities of exposures and confounders to be equal, which that does not always hold. Conclusions The profound and manifold effects of measurement error on estimation and statistical conclusion validity in realistic scenarios indicate that merely mentioning measurement error as a limitation and then dispensing with it is not an adequate response. We also explore questions for optimal study design subject to resource constraints when considering reliability of exposures, covariates, and outcomes.

Multi-dimensional State Space Collapse in Non-complete Resource Pooling Scenarios

We establish an explicit multi-dimensional state space collapse (SSC) for parallel-processing systems with arbitrary compatibility constraints between servers and job types. This breaks major new ground beyond the SSC results and queue length asymptotics in the literature which are largely restricted to complete resource pooling (CRP) scenarios where the steady-state queue length vector concentrates around a line in heavy traffic. The multi-dimensional SSC that we establish reveals heavy-traffic behavior which is also far more tractable than the pre-limit queue length distribution, yet exhibits a fundamentally more intricate structure than in the one-dimensional case.

Unconstraining Multi-Robot Manipulation: Enabling Arbitrary Constraints in ECBS with Bounded Sub-Optimality

Unconstraining Multi-Robot Manipulation: Enabling Arbitrary Constraints in ECBS with Bounded Sub-Optimality

Master–slave elimination scheme for arbitrary smooth nonlinear multi-point constraints

Nonlinear multi-point constraints are essential in modeling various engineering problems, for example in the context of (a) linking individual degrees of freedom of multiple nodes to model nonlinear joints, (b) coupling different element types in finite element analysis, (c) enforcing various types of rigidity in parts of the mesh and (d) considering deformation-dependent Dirichlet boundary conditions. One method for addressing constraints is the master–slave elimination, which offers the benefit of reducing the problem dimension as opposed to Lagrange multipliers and the penalty method. However, the existing master–slave elimination method is limited to linear constraints. In this paper, we introduce a new master–slave elimination method for handling arbitrary smooth nonlinear multi-point constraints in the system of equations of the discretized system. We present a rigorous mathematical derivation of the method. Within this method, new constraints can be easily considered as an item of a “constraint library”; i.e. no case-by-case-programming is required. In addition to the theoretical aspects, we also provide helpful remarks on the efficient implementation. Among others, we show that the new method results in a reduced computational complexity compared to the existing methods. The study also places emphasis on comparing the new approach with existing methods via numerical examples. We have developed innovative benchmarks which encompass all relevant computational properties, and provide analytical and reference solutions. Our findings demonstrate that our new method is as accurate, robust and flexible as the Lagrange multipliers, and more efficient due to the reduction of the total number of degrees of freedom, which is particularly advantageous when a large number of constraints have to be considered.

Multi-dimensional State Space Collapse in Non-complete Resource Pooling Scenarios

The present paper establishes an explicit multi-dimensional state space collapse (SSC) for parallel-processing systems with arbitrary compatibility constraints between servers and job types. This breaks major new ground beyond the SSC results and queue length asymptotics in the literature which are largely restricted to complete resource pooling (CRP) scenarios where the steady-state queue length vector concentrates around a line in heavy traffic. The multi-dimensional SSC that we establish reveals heavy-traffic behavior which is also far more tractable than the pre-limit queue length distribution, yet exhibits a fundamentally more intricate structure than in the one-dimensional case, providing useful insight into the system dynamics. In particular, we prove that the limiting queue length vector lives in a K-dimensional cone of which the set of spanning vectors is random in general, capturing the delicate interplay between the various job types and servers. For a broad class of systems we provide a further simplification which shows that the collection of random cones constitutes a fixed K-dimensional cone, resulting in a K-dimensional SSC. The dimension~K represents the number of critically loaded subsystems, or equivalently, capacity bottlenecks in heavy-traffic, with K=1 corresponding to conventional CRP scenarios. Our approach leverages probability generating function (PGF) expressions for Markovian systems operating under redundancy policies.

Wave interaction with multiple adjacent floating solar panels with arbitrary constraints

The problem of wave interaction with multiple adjacent floating solar panels with arbitrary types and numbers of constraints is considered. All the solar panels are assumed to be homogeneous, with the same physical properties, as well as modeled by using the Kirchhoff-Love plate theory. The motion of the fluid is described by the linear velocity potential theory. The domain decomposition method is employed to obtain the solutions. In particular, the entire fluid domain is divided into two types, the one below the free surface, and the other below elastic plates. The velocity potential in the free surface domain is expressed into a series of eigenfunctions. By contrast, the boundary integral equation and the Green function are employed to construct the velocity potential of fluid beneath the entire elastic cover, with unknowns distributed along two interfaces and jumps of physical parameters of the plates. All these unknowns are solved from the system of linear equations, which is established from the matching conditions of velocity potentials and edge conditions. This approach is confirmed with much higher computational efficiency compared with the one only involving eigenfunction expansion for the fluid beneath each plate. Extensive results and discussions are provided for the reflection and transmission coefficients of water waves, maximum deflection, and principal strain of the elastic plates; especially, the influence of different types and numbers of edge constraints are investigated in detail.

Resource engines

In this paper we aim to push the analogy between thermodynamics and quantum resource theories one step further. Previous inspirations were based predominantly on thermodynamic considerations concerning scenarios with a single heat bath, neglecting an important part of thermodynamics that studies heat engines operating between two baths at different temperatures. Here, we investigate the performance of resource engines, which replace the access to two heat baths at different temperatures with two arbitrary constraints on state transformations. The idea is to imitate the action of a two–stroke heat engine, where the system is sent to two agents (Alice and Bob) in turns, and they can transform it using their constrained sets of free operations. We raise and address several questions, including whether or not a resource engine can generate a full set of quantum operations or all possible state transformations, and how many strokes are needed for that. We also explain how the resource engine picture provides a natural way to fuse two or more resource theories, and we discuss in detail the fusion of two resource theories of thermodynamics with two different temperatures, and two resource theories of coherence with respect to two different bases.

Flan: An Expressive and Efficient Datalog Compiler for Program Analysis

Datalog has gained prominence in program analysis due to its expressiveness and ease of use. Its generic fixpoint resolution algorithm over relational domains simplifies the expression of many complex analyses. The performance and scalability issues of early Datalog approaches have been addressed by tools such as Soufflé through specialized code generation. Still, while pure Datalog is expressive enough to support a wide range of analyses, there is a growing need for extensions to accommodate increasingly complex analyses. This has led to the development of various extensions, such as Flix, Datafun, and Formulog, which enhance Datalog with features like arbitrary lattices and SMT constraints. Most of these extensions recognize the need for full interoperability between Datalog and a full-fledged programming language, a functionality that high-performance systems like Soufflé lack. Specifically, in most cases, they construct languages from scratch with first-class Datalog support, allowing greater flexibility. However, this flexibility often comes at the cost of performance due to the conflicting requirements of prioritizing modularity and abstraction over efficiency. Consequently, achieving both flexibility and compilation to highly-performant specialized code poses a significant challenge. In this work, we reconcile the competing demands of expressiveness and performance with Flan, a Datalog compiler fully embedded in Scala that leverages multi-stage programming to generate specialized code for enhanced performance. Our approach combines the flexibility of Flix with Soufflé’s performance, offering seamless integration with the host language that enables the addition of powerful extensions while generating specialized code for the entire computation. Flan’s simple operator interface allows the addition of an extensive set of features, including arbitrary aggregates, user-defined functions, and lattices, with multiple execution strategies such as binary and multi-way joins, supported by different indexing structures like specialized trees and hash tables, with minimal effort. We evaluate our system on a variety of benchmarks and compare it to established Datalog engines. Our results demonstrate competitive performance and speedups in the range of 1.4× to 12.5× compared to state-of-the-art systems for workloads of practical importance.

Multi-objective optimization for snap-through response of spherical shell panels

Composite spherical shell panels are commonly used for the lightweight design of thin-walled structures in the structure, architecture, aerospace engineering. Depending on the shell panel length, thickness, curvature, and stacking sequence, the load-deflection response under lateral load can be either a completely stable equilibrium path or an unstable snap-through path. In this work, the response of simply supported cross-ply composite spherical shell panels subjected to lateral static loads is studied and optimized. The spherical shell exhibiting snap-through response is optimized to (a) maximize the load required to initiate snap-through and (b) minimize the unstable region. Meanwhile, the spherical shell exhibiting continuous response is optimized to (a) maximize the softening-hardening load and (b) minimize the shell weight. The decision variables for both constraint multi-objective optimization (CMOO) problems include layup sequence, thickness, and the radius of curvature to side length ratio. The nonlinear governing equations are derived based on Kirchhoff–Love hypotheses for thin shells and Hamilton's principle. A novel semi-analytical method is presented based on Bernstein polynomials combined with a fast iterative approach to compute the objective features of the responses. Benefiting from the high efficiency and robustness of the proposed approach, the CMOO problems are solved by the metaheuristic Multi-Objective Artificial Hummingbird Algorithm (MOAHA). Composite spherical shell panels with various geometry and lamination configurations are considered to validate the performance of the proposed method. The optimization of geometric parameters (shell curvature and thickness) in addition to the staking sequence provide a high degree of tailoring of the shell performance to meet various objectives. These parameters are taken as design variables that must be optimized to achieve some objectives and satisfy arbitrary nonlinear constraints.

A material–structure-wave integrated hydro-viscoelastic model for floating composite structure with generalized boundary

A floating viscoelastic composite structure with generalized elastic constraints, in which the material–structure-wave interaction induces energy dissipation, is analyzed with a novel cross-scale integrated hydro-viscoelastic model. A double-level three-dimensional (3D) representative volume element model coupling mesoscale reinforced particles to a macroscale composite structure is established to obtain the structure's viscoelastic dynamic equation. Therefore, a hydro-viscoelasticity analytical model for floating composite structures with arbitrary elastic constraints under wave action is developed in the context of potential flow theory. In the process of solving the velocity potential, the dispersion equation of the water covered by the composite structure is derived. The hydrodynamic and mechanical behaviors of the floating composite structure, including the reflection coefficient, transmission coefficient, dissipation coefficient, deflection, shear force, and bending moment, are comprehensively calculated, which depend on the cross-scale integrated effects of wave action, structural features, material characteristics, and constraint combinations. This material–structure-wave integrated model would be useful to elucidate the hydro-viscoelastic dissipation mechanism of marine structures.

Pliability and Approximating Max-CSPs

We identify a sufficient condition, treewidth-pliability , that gives a polynomial-time algorithm for an arbitrarily good approximation of the optimal value in a large class of Max-2-CSPs parameterised by the class of allowed constraint graphs (with arbitrary constraints on an unbounded alphabet). Our result applies more generally to the maximum homomorphism problem between two rational-valued structures. The condition unifies the two main approaches for designing a polynomial-time approximation scheme. One is Baker’s layering technique, which applies to sparse graphs such as planar or excluded-minor graphs. The other is based on Szemerédi’s regularity lemma and applies to dense graphs. We extend the applicability of both techniques to new classes of Max-CSPs. However, we prove that the condition cannot be used to find solutions (as opposed to approximating the optimal value) in general. Treewidth-pliability turns out to be a robust notion that can be defined in several equivalent ways, including characterisations via size, treedepth, or the Hadwiger number. We show connections to the notions of fractional-treewidth-fragility from structural graph theory, hyperfiniteness from the area of property testing, and regularity partitions from the theory of dense graph limits. These may be of independent interest. In particular, we show that a monotone class of graphs is hyperfinite if and only if it is fractionally-treewidth-fragile and has bounded degree.