Wil Schilders

• Eindhoven University of Technology

Structural dynamics models with nonlinear stiffness appear, for example, when analyzing systems with nonlinear material behavior or undergoing large deformations. For complex systems, these models become too large for real-time applications or multi-query workflows. Hence, model reduction is needed. However, the mathematical operators of these mode...

Optimization of transient models is required in several domains related to thermo-mechanical reliability of electronics, such as Prognostic Health Monitoring (PHM) and design optimization. A novel framework for efficient (local) parameter optimization of transient models in the \(\mathscr {H}_2\) norm is proposed. The optimization is feasible for l...

The work of this paper focuses on model order reduction for a special class of nonlinear dynamical systems, that is, the class of quadratic-bilinear dynamical systems. This kind of systems can be used to represent other nonlinear dynamical systems with strong nonlinearities such as exponent and high-order polynomials. This paper addresses the H2 op...

This paper proposes a data-based approach for model order reduction that preserves incremental stability properties. Existing data-based approaches do typically not preserve such incremental system properties, especially for nonlinear systems. As a result, instability of the constructed model commonly occurs for inputs outside the training set, whi...

This is the first chapter of a three-volume series dedicated to theory and application of Model Order Reduction (MOR). We motivate and introduce the basic concepts and notation, with reference to the two main cultural approaches to MOR: the system-theoretic approach employing state-space models and transfer function concepts (Volume 1), and the num...

Two-phase flows are frequently modelled and simulated using the Two-Fluid Model (TFM) and the Drift Flux Model (DFM). This paper proposes Stokes–Dirac structures with respect to which port-Hamiltonian representations for such two-phase flow models can be obtained. We introduce a non-quadratic candidate Hamiltonian function and present dissipative H...

Many single- and multi-phase fluid dynamical systems are governed by non-linear evolutionary equations. A key aspect of these systems is that the fluid typically flows across spatially and temporally varying cross-sections. We, first, show that not any choice of state-variables may be apt for obtaining a port-Hamiltonian realization under spatially...

We present a structure-preserving spatial discretization method for infinite-dimensional non-linear port-Hamiltonian representations of a commonly used one-dimensional two-phase flow model: the Two-Fluid Model. We introduce the port-Hamiltonian representation of this two-phase flow model and then invoke a mixed-finite-element method to perform a st...

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This three-volume handbook covers methods as well as applications. This third volume focuses on applications in engineering, biomedical engineering,...

This chapter has two main objectives: first, to propose a computer-aided consistent and accurate description of the behavior of electromagnetic devices at various speeds or frequencies and, second, to describe procedures to generate compact electrical circuits for them, with an approximatively equivalent behavior. The extracted models should have a...

In this paper, approximate well-balanced finite-volume schemes are developed for the isothermal Euler equations and the drift flux model, widely used for the simulation of single-and two-phase flows. The proposed schemes, which are extensions of classical schemes, effectively enforce the well-balanced property to obtain a higher accuracy compared t...

The yield of an Integrated Circuit (IC) is commonly expressed as the fraction (in %) of working chips overall manufactured chips and often interpreted as the failure probability of its analog blocks. We consider the Importance Sampling Monte Carlo (ISMC)Importance Sampling Monte Carlo as a reference method for estimating failure probabilities. For...

The main motivation of this work is lying in the acceleration of transient simulation of Analog Mixed Signal circuits. In the electronics industry, smaller and faster electronic devices are always demanded. Full device-parasitic transient simulations of realistic circuits are time consuming or even infeasible due to a huge number of electrical comp...

We propose a new model order reduction (MOR) approach to obtain effective reduction for transport-dominated problems or hyperbolic partial differential equations. The main ingredient is a novel decomposition of the solution into a function that tracks the evolving discontinuity and a residual part that is devoid of shock features. This decompositio...

Automation of managed pressure drilling (MPD) enables fast and accurate pressure control in drilling operations. The performance that can be achieved by automated MPD is determined by, first, the controller design and, second, the hydraulics model that is used as a basis for controller design. On the one hand, such hydraulics model should be able t...

Automated managed-pressure drilling (MPD) is a method to enhance downhole pressure-control performance and safety during drilling operations. It is becoming more common to use model-based simulation for the evaluation of pressure-control systems designed for MPD automation before using those in the field. This demands a representative hydraulics-si...

To circumvent restrictions of conventional drilling methods, such as slow control actions and inability to drill depleted reservoirs, a drilling method called managed pressure drilling (MPD) has been developed. In MPD, single‐phase flow processes can be modeled as a feedback interconnection of a high‐order linear system and a low‐order nonlinear sy...

This work focuses on the model order reduction problem for bilinear control systems with nonzero initial conditions. Based on the Volterra series analysis, the system response can be decomposed into three parts. The first two parts are the zero input response and zero initial condition response of the system. The third part describes the response w...

This paper presents a model reduction approach for systems of hyperbolic partial differential equations (PDEs) with nonlinear boundary conditions. These systems can be decomposed into a feedback interconnection of a linear hyperbolic subsystem and a static nonlinear mapping. This structure motivates us to reduce the overall model complexity by only...

Burgers’ equation is a nonlinear scalar partial differential equation, commonly used as a testbed for model order reduction techniques and error estimates. Model order reduction of the parameterized Burgers’ equation is commonly done by using the reduced basis method. In this method, an error estimate plays a crucial rule in both accelerating the o...

In this paper a survey is presented of the use of finite element methods for the simulation of the behaviour of semiconductor devices. Both standard and mixed finite element methods are considered. We indicate how the various mathematical models of semiconductor device behaviour can be obtained from the Boltzmann transport equation and the appropri...

Many physical phenomena, such as mass transport and heat transfer, are modeled by systems of partial differential equations with time-varying and nonlinear boundary conditions. Control inputs and disturbances typically affect the system dynamics at the boundaries and a correct numerical implementation of boundary conditions is therefore crucial. Ho...

This paper presents a modification of a classical Godunov-type scheme for the numerical simulation of a two-phase flow in a pipe with a piecewise constant cross-sectional area. This type of flow can occur in wellbores during drilling for oil and gas as well as after well completion. Contrary to classical finite-volume schemes, the numerical scheme...

This paper focuses on the model reduction problem for a special class of linear parameter-varying systems. This kind of systems can be reformulated as bilinear dynamical systems. Based on the bilinear system theory, we give a definition of the \(\mathcal {H}_{2}\) norm in the generalized frequency domain. Then, a model reduction method is proposed...

The dimension of transistors shrinks with each new technology developed in the semiconductor industry. The extreme scaling of transistors introduces important statistical variations in their process parameters. A large digital integrated circuit consists of a very large number (in millions or billions) of transistors, and therefore the number of st...

We present a numerical technique that automatically identifies a suitable initial solution to start periodic steady-state methods for simulating non-autonomous circuits at transistor-level. The method avoids the guessing of the initial solution, which may result in divergence of the steady-state method used. For high-Q oscillating circuits, acceler...

This paper presents a drop-threshold incomplete LD⁻¹LT (δ) factorization constraint preconditioner for saddle-point systems using a threshold parameter δ. A transformed saddle-point matrix is partitioned into a block structure with blocks of order 1 and 2 constituting ‘a priori pivots’. Based on these pivots an incomplete LD⁻¹LT (δ) factorization c...

Automated Managed Pressure Drilling (MPD) is a method for fast and accurate pressure control in drilling operations. The achievable performance of automated MPD is limited, firstly, by the control system and, secondly, by the hydraulics model based on which this control system is designed. Hence, an accurate hydraulics model is needed that, at the...

In this article, we simulate traveling liquid slugs in conduits, as they may occur in systems carrying high-pressure steam. We consider both horizontal and inclined pipes in which the slug is accelerated by a suddenly applied pressure gradient, while at the same time, gravity and friction work in the opposite direction. This causes a steep slug fro...

In this article, we simulate traveling liquid slugs in conduits, as they may occur in systems carrying high-pressure steam. We consider both horizontal and inclined pipes in which the slug is accelerated by a suddenly applied pressure gradient, while at the same time, gravity and friction work in the opposite direction. This causes a steep slug fro...

We consider the Importance Sampling Monte Carlo (ISMC) as a reference probability estimator for estimating very small probabilities in the context of analog circuits design. We propose a surrogate based hybrid ISMC method to accelerate the estimation of probabilities when the budget of simulations is limited. The Kriging model is used as a surrogat...

We present a methodology to simulate industrial integer-$N$ phase-locked loops (PLLs) at a verification level, as accurate as and faster than transistor-level simulation. The accuracy is measured on the PLL factors of interest, i.e., locking time, power consumption, phase noise and jitter (period and long-term). The speedup factor tends to the divi...

We compare several approximations for second derivatives with Smoothed Particle Hydrodynamics (SPH). A first-order consistent approximation, derived from the zeroth-order consistent Corrective Smoothed Particle Method (CSPM), is proposed. The accuracy of the new method (ICSPM) is similar to that of the Finite Particle Method (FPM) and Modified Smoo...

Imperfections in manufacturing processes may cause unwanted connections (faults) that are added to the nominal, “golden”, design of an electronic circuit. By fault simulation we simulate all situations: new connections and each with different values for the newly added element. We also consider “opens” (broken connections). During the transient sim...

We present a method to speed up noise-free and noisy time domain simulations of industrial integer-N PLLs, while extracting the main factors of interest which circuit designers are interested in, i.e., locking time, power consumption, phase noise and jitter, within desirable error levels. The procedure is based on oscillator's sensitivity analysis...

In this book, two MOR methods for linear constant coefficient DAEs are discussed. These MOR methods are: the Index-aware MOR (IMOR) and Implicit IMOR (IIMOR) methods. They both reduce DAEs of any index by first decoupling it into differential and algebraic parts using projector, matrix and basis chain.KeywordsImplicit Image (IIMOR)Basic ChainAlgebr...

In this chapter, we illustrate the robustness of the IMOR method on large scale problems from real-life applications.

In this chapter, we introduce the differential algebraic equations which we abbreviate as DAEs. DAEs arise in a variety of applications such as modelling constrained multibody systems, electrical networks, aerospace engineering, chemical processes, computational fluid dynamics, gas transport networks, see [10–12, 35]. Therefore their analysis and n...

In this chapter, we discuss how to decouple DAEs using matrix, projector and basis chains. This approach is based on the projector and matrix chains introduced in [25].

In this chapter, we discuss the index-aware model order reduction (IMOR) and its invariant the implicit-IMOR(IIMOR) method. We use the decoupled systems (3. 2. 11) and (3. 7. 1) to derive the IMOR and IIMOR method respectively.

Many SPH approximations for second-order derivatives, or the Laplacian, suffer from the presence of boundaries and irregularities in the particle distribution. In this paper we discuss four estimates to the Laplacian: the Brookshaw approximation, CSPM, MSPH and ICSPM – the latter of which is derived in this work. We theoretically derive the converg...

This chapter contains three advanced topics in model order reduction (MOR): nonlinear MOR, MOR for multi-terminals (or multi-ports) and finally an application in deriving a nonlinear macromodel covering phase shift when coupling oscillators. The sections are offered in a preferred order for reading, but can be read independently. Section 6.1, writt...

This Chapter introduces parameterized, or parametric, Model Order Reduction (pMOR). The Sections are offered in a prefered order for reading, but can be read independently. Section 5.1, written by Jorge Fernández Villena, L. Miguel Silveira, Wil H.A. Schilders, Gabriela Ciuprina, Daniel Ioan and Sebastian Kula, overviews the basic principles for pM...

There exists many Model Order Reduction (MOR) methods for ODEs but little had been done to reduce DAEs especially higher index DAEs. In principle, if the matrix pencil of a DAE is regular, it is possible to use conventional MOR techniques to obtain reduced order models, which are generally ODEs. However, as far as their numerical treatment is conce...

Schilders' factorization can be used as a basis for preconditioning indefinite linear systems which arise in many problems like least-squares, saddle-point and electronic circuit simulations. Here we consider its application to resistor network modeling. In that case the sparsity of the matrix blocks in Schilders' factorization depends on the spars...

Purpose – Model order reduction (MOR) has been widely used in the electric networks but little has been done to reduce higher index differential algebraic equations (DAEs). The paper aims to discuss these issues. Design/methodology/approach – Most methods first do an index reduction before reducing a higher DAE but this can lead to a loss of physi...

The factorization method presented in this paper takes advantage of the special structures and properties of saddle point matrices. A variant of Gaussian elimination equivalent to the Cholesky's factorization is suggested and implemented for factorizing the saddle point matrices block-wise with small blocks of order 1 and 2. The Gaussian eliminatio...

In this paper, we give an overview of the development of industrial mathematics in Europe. The advent of activities is in the 1970s, when, especially in Oxford, the potential of applications of mathematics was realized by Alan Tayler and co-workers, and the very successful study groups with industry were started. It led to discussions about Europea...

Purpose – Imperfections in manufacturing processes may cause unwanted connections (faults) that are added to the nominal, “golden”, design of an electronic circuit. By fault simulation one simulates all situations. Normally this leads to a large list of simulations in which for each defect a steady-state (direct current (DC)) solution is determined...

In this paper we present two remedies for particle clustering in SPH. Since particle clustering is the consequence of a diminishing kernel gradient for small inter-particle distances, the first method uses a convex kernel with a non-zero kernel gradient at the origin. The second method is based on inter-particle collisions. They are both compared w...

In this paper we present two remedies for particle clustering in SPH. Since particle clustering is the consequence of a diminishing kernel gradient for small inter-particle distances, the first method uses a convex kernel with a non-zero kernel gradient at the origin. The second method is based on inter-particle collisions. They are both compared w...

This paper describes an original methodology for the modeling of parasitic inductive couplings. The key idea is the use of magnetic hooks which are gates for magnetic fluxes that cross conductive loops and consequently induce parasitic voltages, thus disturbing the signal integrity. The multiple connected domains of integrated circuits are modeled...

We consider Uncertainty Quantification (UQ) by expanding the solution in so-called generalized Polynomial Chaos expansions. In these expansions the solution is decomposed into a series with orthogonal polynomials in which the parameter dependency becomes an argument of the orthogonal polynomial basis functions. The time and space dependency remains...

This paper introduces the implicit-IMOR method for differential algebraic equations. This method is a modification of the Index-aware model order reduction (IMOR) method proposed in our earlier papers which is the explicit-IMOR method. It also involves first splitting the differential-algebraic equations (DAEs) into differential and algebraic parts...

Many different methods have been devised to solve the nonlinear systems of equations that model water distribution networks. Probably the most popular is Todini and Pilati’s global gradient algorithm (GGA). Given the GGA’s success, alternative methods have not aroused much interest. One example is the co-tree method, which requires some cumbersome...

Model order reduction (MOR) has become an important tool in the design of complex high-tech systems. It can be used to find a low-order model that approximates the behavior of the original high-order model, where this low-order approximation facilitates both the computationally efficient analysis and controller design for the system to induce desir...

Many different methods have been devised to solve the nonlinear systems of equations that model water distribution networks. Probably the most popular is Todini and Pilati’s global gradient algorithm (GGA). Given the GGA’s success, alternative methods have not aroused much interest. One example is the co-tree method, which requires some cumbersome...

In this paper, popular model reduction techniques from the fields of structural dynamics, numerical mathematics and systems and control are reviewed and compared. The motivation for such a comparison stems from the fact that the model reduction techniques in these fields have been developed fairly independently. In addition, the insight obtained by...

The interconnect layouts of chips can be modeled by large resistor networks. In order to be able to speed up simulations of such large networks, reduction techniques are applied to reduce the size of the networks. For some class of networks, an existing reduction strategy does not provide sufficient reduction in terms of the number of resistors app...

We introduce a model order reduction (MOR) procedure for differential-algebraic equations, which is based on the intrinsic differential equation contained in the starting system and on the remaining algebraic constraints. The decoupling procedure in differential and algebraic part is based on the projector and matrix chain which leads to the defini...

To solve (partial) differential equations it is necessary to have good numerical approximations. In SPH, most approximations suffer from the presence of boundaries. In this work a new approximation for the second-order derivative is derived and numerically compared with two other approximation methods for a simple test case. The new method is sligh...

A model order reduction method for index-2 differential-algebraic equations (DAEs) is introduced, which is based on the intrinsic differential equations and on the remaining algebraic constraints. This extends the method introduced in a previous paper for index-1 DAEs. This procedure is implemented numerically and the results show numerical evidenc...

In this paper, we propose a semianalytical solution for evaluation of field distributions around a short rectangular crack in a metallic half-space excited by long current-carrying wires with arbitrary frequency. The governing Helmholtz equation is solved in three dimensions by separation of variables. The solution is obtained by developing 2-D Fou...

Electromagnetic descriptor models are models which lead to differential algebraic equations (DAEs). Some of these models mostly arise from electric circuit and power networks. The most frequently used modeling technique in the electric network design is the modified nodal analysis (MNA) which leads to differential algebraic equations in descriptor...

Purpose To simulate large parasitic resistive networks, one must reduce the size of the circuit models through methods that are accurate and preserve terminal connectivity and network sparsity. The purpose here is to present such a method, which exploits concepts from graph theory in a systematic fashion. Design/methodology/approach The model orde...

Simulation of the influence of interconnect structures and substrates is essential for a good understanding of modern chip behavior. Sometimes such simulations are not feasible with current circuit simulators. We propose an approach to reduce the large resistor networks obtained from extraction of the parasitic effects that builds upon the work in...

This paper presents a computationally efficient model order reduction (MOR) technique for interconnected systems. This MOR technique preserves block structures and zero blocks and exploits separate MOR approximations for the individual sub-systems in combination with low rank approximations for the interconnection blocks. The reduction is demonstra...

A novel model order reduction (MOR) method, SparseRC, for multiterminal RC circuits is proposed. Specifically tailored to systems with many terminals, SparseRC employs graph-partitioning and fill-in reducing orderings to improve sparsity during model reduction, while maintaining accuracy via moment matching. The reduced models are easily converted...

In this paper, we present a new gauge technique for the Newton Raphson method to solve the periodic steady state (PSS) analysis of free-running oscillators in the time domain. To find the frequency a new equation is added to the system of equations. Our equation combines a generalized eigenvector with the time derivative of the solution. It is dyna...

This research project is concerned with the modelling of the electric charge phenomena that spacecrafts are subject to. These phenomena are sources of in-orbit failures since high potential differences induce the formation of electric arcing which can produce irreversible failures on on-board devices. The numerical simulation of such phenomena, tak...

In 2006 Volvo Car Corporation initiated together with the Swedish automotive industry and Vinnova a research project on developing software for simulation of processes in automotive paint shops. The software development was done by the Fraunhofer-Chalmers Research Centre for Industrial Mathematics (FCC) and the first version of the virtual spray pa...

The challenge is to select the portfolio of financial securities which is optimal under given criteria.

Monolix: a powerful tool for population pharmacology

Helsinki University of Technology1, University of Jyväskylä and Numerola Oy have co-operated in a research project which investigates the numerical modelling of bentonite buffer in nuclear waste management.

Three priorities should be the heart of Europe 20202: • Smart growth, by developing an economy based on knowledge and innovation. • Sustainable growth, by promoting a more resource efficient, greener and more competitive economy. • Inclusive growth, by fostering a high-employment economy delivering economic, social and territorial cohesion. These t...

The project concerns new optimization methods for intensity-modulated radiation therapy.

This paper deals with reformulating the electro- magnetic field equations for a combined EM and TCAD approach in such a way that both extreme high and low frequencies can be solved. The importance of the method is found in eliminating the need for direct solvers which are restricted in application to very large systems. We elaborate on the numerica...

In this paper, we discuss the present and future needs of the electronics industry with regard to model order reduction. The industry has always been one of the main motivating fields for the development of MOR techniques, and continues to play this role. We discuss the search for provably passive methods, as well as passivity enforcement methods t...

This unique book presents real world success stories of collaboration between mathematicians and industrial partners, showcasing first-hand case studies, and lessons learned from the experiences, technologies, and business challenges that led to the successful development of industrial solutions based on mathematics. These success stories show the...

Electro Static Discharge (ESD) analysis is of vital importance during the design of large-scale integrated circuits, since it gives insight in how well the interconnect can handle unintended peak charges. Due to the increasing amount of interconnect and metal layers, ESD analysis may become very time consuming or even unfeasible. We propose an algo...

The super node algorithm performs model order reduction based on physical principles. Although the algorithm provides us with compact models, its stability and passivity have not thoroughly been studied yet. The loss of passivity is a serious problem because simulations of the reduced network may encounter artificial behavior which render the simul...

This part addresses the challenging topic of solving coupled problems. The increasing necessity to solve complex problems in the science and engineering community, accounting for all the coupling occurring at the different scales of the problem, requires the development of new ideas and methods which can effectively provide accurate numerical solut...

This paper describes a locally one-dimensional finite-difference time domain method for the two-dimensional time-dependent simulation of semiconductor devices. This approach leads to significant reduction of the semiconductor simulation time. We can reach over 80% reduction in the simulation time by using this technique while maintaining the same d...

The super node algorithm performs model order reduction based on physical principles. Although the algorithm provides us with compact models, its passivity has not thoroughly been studied yet. The loss of passivity is a serious problem because simulations of the reduced network may encounter artificial behavior which render the simulations useless....

Large resistor networks arise during the design of very-large-scale integration chips as a result of parasitic extraction and electro static discharge analysis. Simulating these large parasitic resistor networks is of vital importance, since it gives an insight into the functional and physical performance of the chip. However, due to the increasing...

For a fast simulation of interconnect structures we consider preconditioned iterative solution methods for large complex valued linear systems. In many applications the discretized equations result in ill-conditioned matrices, and efficient preconditioners are indispensable to solve the linear systems accurately. We apply the dual threshold incompl...

In this paper we discuss the use of the state-space modelling MOESP algorithm to generate precise information about the number of neurons and hidden layers in dynamic neural networks developed for the behavioural modelling of electronic circuits. The Bartels–Stewart algorithm is used to transform the information from the MOESP algorithm to the neur...

Design of integrated RF circuits requires detailed insight in the behavior of the used components. Unintended coupling and perturbation effects need to be accounted for before production, but full simulation of these effects can be expensive or infeasible. In this paper, we present a method to build nonlinear phase macromodels of voltage-controlled...