Simplification of the Isotropic Generalized Stop-Type Prandtl-Ishlinskii Vector Hysteresis Operator

Jul 16, 2026

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Thermodynamically formulated generalized Prandtl–Ishlinskii stop-type operators model hysteresis nonlinearities but require local iterative Newton updates to compute each hysteron’s internal memory state, creating high computational cost in finite element simulations. A simplified formulation replaces the nonlinear mapping on the stop operator with an identity, directly weighting hysteron outputs while fully preserving the nonlinear anhysteretic response using ramp dead-zone basis functions. For isotropic cases, analytical return-point mapping yields a closed-form local plastic correction, removing per-hysteron iterations. Integrated into an FE solver, the approach reduces computation time while matching generalized-model accuracy.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages4

Reading Time10 min

Introduction

Simplified model and its formulation

Operator simplification

Anhysteretic characteristic

Hysteretic formulation

Thermodynamic consistency

Parameter identification

Finite element simulation and results

Conclusion

Simplification of the Isotropic Generalized Stop-Type Prandtl-Ishlinskii Vector Hysteresis Operator Using Analytical Return-Point Mapping Arvinth Shankar 1,2 , Klaus Kuhnen 1 , Iryna Kulchytska-Ruchka 1 , Sebastian Schps2 1Robert Bosch GmbH, Bosch Research and Advance Engineering, Germany 2 Computational Electromagnetics Group, Technische Universitt Darmstadt, Germany While the thermodynamically formulated generalized Prandtl–Ishlinskii stop-type operator effectively captures hysteresis nonlinearities, it requires a local iterative procedure to update each hysteron, resulting in considerable computational effort. In this work, we propose a simplified thermodynamic formulation of the generalized Prandtl–Ishlinskii stop operator. The nonlinear mapping on the stop operator is replaced by an identity, such that the hysteresis operators are directly weighted through their outputs, while the nonlinear anhysteretic response, represented by ramp dead-zone basis functions, is fully preserved. For isotropic cases, this simplification enables a closed-form solution for the local plastic correction, eliminating per-hysteron iterative Newton updates. The resulting constitutive mapping is integrated into a finite element solver, and numerical results show a significant reduction in computation time with accuracy comparable to the generalized model. Index Terms—computational electromagnetism, vector hysteresis, constitutive relation, Prandtl–Ishlinskii model arXiv :2607 .07575v 1 [ cs .CE] 8 Jul 2026 I. INTRODUCTION REAL ferromagnetic materials exhibit complex hysteresis effects, requiring memory-dependent constitutive mapping between the magnetic field H and the flux density B for solving low-frequency Maxwell’s equations. Operator-based phenomenological approaches [1] are widely used, expressed as a vector-valued, rate-independent operator H = V [B] . The generalized Prandtl–Ishlinskii model [2] effectively characterizes hysteresis nonlinearities and its stop-type formulation defines the mapping V through a weighted superposition of generalized stop operators. Its rheological representation [3],[4] combines a nonlinear elastic spring in parallel with multiple generalized vector-stop hysterons, and a thermodynamic interpretation of such nonlinear vector-stop elements for anisotropic materials is discussed in [5] . Thermodynamically formulated models provide physically meaningful constructions and allows decomposition of dissipative and reactive power [3] . In the thermodynamically formulated generalized Prandtl– Ishlinskii stop-type model [5], each hysteron is split into an elastic (reversible) part and a plastic (irreversible) part, where the plastic part constitutes the internal memory state. The hysteron output is a nonlinear function of the elastic part, yielding an implicit update equation for each hysteron and can only be resolved iteratively through the Newton method. This local, nonlinear, iterative update incurs considerable computational effort, particularly in finite element (FE) simulations. Although V is local and parallelizable across integration points, the per-point hysteretic updates remain expensive. In this work, we propose a simplification of the generalized Prandtl–Ishlinskii stop operator within the thermodynamic framework, while retaining the essential hysteresis properties. This simplification linearly weights the stop-operator outputs, and for isotropic cases, this yields an analytical returnpoint mapping for the local hysteron updates, eliminating the need for iterative Newton procedures. Integrated into an FE framework, the simplified constitutive mapping enables faster evaluation of V [B] at each integration point, reducing the overall computational cost. The paper is organized as follows. In Sec. II-A, we introduce the operator simplification and its rheological interpretation. The anhysteretic and hysteretic formulations are presented in Sec. II-B and Sec. II-C, respectively. Thermodyn

FAQs

Why does the generalized thermodynamic Prandtl–Ishlinskii stop-type operator require iterative computation?

Each hysteron’s implicit update depends on its internal elastic and plastic parts, and the local nonlinear update equation can only be resolved iteratively using Newton’s method.

What is the key idea behind the proposed simplification of the stop-type operator?

The nonlinear mapping applied to the stop operator is replaced by an identity, so the hysteresis contribution is obtained by directly weighting the stop-operator outputs, while the anhysteretic nonlinear response is preserved.

What performance benefit does the simplified model achieve when integrated into a finite element solver?

For isotropic cases, analytical return-point mapping provides a closed-form local plastic correction, eliminating per-hysteron iterative Newton updates and significantly reducing computation time while maintaining comparable accuracy.

Jul 16, 2026

Upload by Asher