In this thesis, a numerical approach based on a potential flow method has been developed in order to simulate unsteady rotor/stator interaction, and to predict the unsteady performance of a propeller and its rudder. The method is first developed and tested in two-dimensions by using a boundary element method in which a front hydrofoil is moving downward, while a back hydrofoil is stationary. The wakes of the two hydrofoils are modeled by continuous dipole sheets and determined in time by applying a force free-condition on each wake surface. The wake/hydrofoil interaction is de-singularized by applying a numerical fence on the surface of the back hydrofoil. The viscous wake/hydrofoil interaction is considered by employing a viscous wake vorticity model on the rotor's wake surface. The present method is validated by comparison with analytical solutions, experimental data and by using the results from a commercial Reynolds Averaged Navier-Stokes (RANS) solver for the same set-up and conditions. The numerical approach is further extended to three-dimensions to predict the mutual interaction between a propeller and rudder. A fully unsteady wake alignment algorithm is implemented into a Vortex Lattice Method to simulate the unsteady propeller flow. The interaction between propeller and rudder is investigated in a fully unsteady manner, where a panel method is used to solve the flow around the rudder, and a vortex lattice method is used to solve the flow around the propeller. The interaction between a propeller and its rudder is considered in an iterative manner by solving the propeller and the rudder problems separately and by including the unsteady effects of one component on the other. The effect of the unsteady propeller-rudder interaction on the mean and on the unsteady propeller/rudder performance, including sheet cavitation on the rudder, is studied. / text

Au cours de cette thèse, un modèle potentiel résolvant les équations d’Euler-Zakharov a été développé dans le but de simuler la propagation de vagues et d’états de mer irréguliers et multi-directionnels, du large jusqu’à la côte, sur des bathymétries variables. L’objectif est de représenter les effets non-linéaires et dispersifs le plus précisément possible pour des domainescôtiers bidimensionnels (dans le plan horizontal) de l’ordre de quelques kilomètres.La version 1DH initiale du modèle, résolvant le problème aux limites de Laplace à l’aide de schémas aux différences finies d’ordre élevé dans la direction horizontale combinés à une approche spectrale sur la verticale, a été améliorée et validée. L’implémentation de conditions aux limites de type Dirichlet et Neumann pour générer des vagues dans le domaine a été étudiée en détail. Dans la pratique, une zone de relaxation a été utilisée en complément deces conditions pour améliorer la stabilité du modèle.L’expression analytique de la relation de dispersion a été établie dans le cas d’un fond plat. Son analyse a montré que la représentation des effets dispersifs s’améliorait significativement avec l’augmentation de la résolution sur la direction verticale (i.e. avec le degré maximal de la basede polynômes de Tchebyshev utilisée pour projeter le potentiel des vitesses sur la verticale).Une étude de convergence menée pour des ondes solitaires modérément à fortement non-linéaires a confirmé la convergence exponentielle avec la résolution verticale grâce à l’approche spectrale, ainsi que les convergences algébriques en temps et en espace sur l’horizontale avec des ordres d’environ 4 (ou plus) en accord avec les schémas numériques utilisés.La comparaison des résultats du modèle à plusieurs jeux de données expérimentales a démontré les capacités du modèle à représenter les effets non-linéaires induits par les variations de bathymétrie, notamment les transferts d’énergie entre les composantes harmoniques, ainsi que la représentation précise des propriétés dispersives. Une formulation visco-potentielle a également été implémentée afin de prendre en compte les effets visqueux induits par la dissipation interne et le frottement sur le fond. Cette formulation a été validée dans le cas d’une faible viscosité avec un fond plat ou présentant une faible pente.Dans le but de représenter des champs de vagues 2DH, le modèle a été étendu en utilisant une discrétisation non-structurée (par nuage de points) du plan horizontal. Les dérivées horizontales ont été estimées à l’aide de la méthode RBF-FD (Radial Basis Function-Finite Difference), en conservant l’approche spectrale sur la verticale. Une étude numérique de sensibilité a été menée afin d’évaluer la robustesse de la méthode RBF-FD, en comparant différents types de RBFs, avec ou sans paramètre de forme et l’ajout éventuel d’un polynôme. La version 2DH du modèle a été utilisée pour simuler deux expériences en bassin, validant ainsi l’approche choisie et démontrant son applicabilité pour simuler la propagation 3D des vagues faisant intervenir des effets non-linéaires. Dans le but de réduire le temps de calcul et de pouvoir appliquer le code à des simulations sur de grands domaines, le code a été modifié pour utiliser le solveur linéaire direct en mode parallèle / In this work, a potential flow model based on the Euler-Zakharov equations was developed with the objective of simulating the propagation of irregular and multidirectional sea states from deep water conditions to the coast over variable bathymetry. A highly accurate representation of nonlinear and dispersive effects for bidimensional (2DH) nearshore and coastal domains on the order of kilometers is targeted.The preexisting 1DH version of the model, resolving the Laplace Boundary Value problem using a combination of high-order finite difference schemes in the horizontal direction and a spectral approach in the vertical direction, was improved and validated. The generation of incident waves through the implementation of specific Dirichlet and Neumann boundary conditions was studied in detail. In practice, these conditions were used in combination witha relaxation zone to improve the stability of the model.The linear dispersion relation of the model was derived analytically for the flat bottom case. Its analysis showed that the accuracy of the representation of dispersive effects improves significantly by increasing the vertical resolution (i.e. the maximum degree of the Chebyshev polynomial basis used to project the potential in the vertical). A convergence study conducted for moderate to highly nonlinear solitary waves confirmed the exponential convergence in the vertical dimension owing to the spectral approach, and the algebraic convergence in time and in space (horizontal dimension) with orders of about 4 (or higher) in agreement with the numerical schemes used.The capability of the model to represent nonlinear effects induced by variable bathymetry, such as the transfer of energy between harmonic components, as well as the accurate representation of dispersive properties, were demonstrated with comparisons to several experimental data sets. A visco-potential flow formulation was also implemented to take into account viscous effects induced by bulk viscosity and bottom friction. This formulation was validated inthe limit of small viscosity for mild slope bathymetries.To represent 2DH wave fields in complex nearshore domains, the model was extended using an unstructured discretization (scattered nodes) in the horizontal plane. The horizontal derivatives were estimated using the RBF-FD (Radial Basis Function - Finite Difference) method, while the spectral approach in the vertical remained unchanged. A series of sensitivity tests were conducted to evaluate numerically the robustness of the RBF-FD method, including a comparison of a variety of RBFs with or without shape factors and augmented polynomials. The 2DH version of the model was used to simulate two wave basin experiments, validating the approach and demonstrating the applicability of this method for 3D wave propagation, including nonlinear effects. As an initial attempt to improve the computational efficiency ofthe model for running simulations of large spatial domains, the code was adapted to use a parallelized direct linear solver

Le calcul des efforts hydrodynamiques de premier ordre sur un ou plusieurs corps perçant la surface libre est aujourd'hui bien maîtrisé, et plusieurs codes de calcul implémentant la méthode des singularités (dite BEM ou méthode d'élément frontière) ont été développés. Le cadre est la théorie linéarisée des écoulements potentiels à une surface libre. Dans ces codes BEM, les singularités utilisées ont la propriété intrinsèque de satisfaire à la fois l'équation de Laplace dans le domaine fluide ainsi que la condition linéarisée de surface libre. Ces singularités, dites fonctions de Green à surface libre, dans le domaine fréquentiel en profondeur infinie et sans vitesse d'avance constituent le point focal de cette thèse. Tout d'abord, les expressions mathématiques existantes pour la fonction de Green de surface libre sont examinées. Douze expressions différentes sont passées en revue et analysées. Plusieurs méthodes numériques existantes sont comparées par rapport à leur temps de calcul et leur précision. Ensuite, une série d'équations différentielles ordinaires (ODEs) pour les fonctions de Green de surface libre dans le domaine temporel et le domaine fréquentiel et leur gradient est établie. Ces ODEs peuvent être utilisées pour mieux comprendre les propriétés de la fonction de Green et peuvent constituer un moyen alternatif de calculer ces fonctions de Green et leurs dérivées. Cependant, il est difficile de résoudre numériquement ces ODEs à cause de l'existence d'une singularité à l'origine. Cette difficulté est éliminée en modifiant les ODEs par l'utilisation de nouvelles fonctions sans singularité. Les nouvelles ODEs sont ensuite écrites sous forme canonique en utilisant une nouvelle définition de la fonction vectorielle. La forme canonique peut être résolue avec les conditions initiales à l'origine puisque tous les termes impliqués sont finis. Une méthode d'expansion basée sur une série de fonctions logarithmiques et de polynômes ordinaires, très efficace pour les problèmes de basse fréquence, a également été développée pour obtenir des solutions analytiques. Enfin, la méthode basée sur les ODE pour calculer la fonction de Green est implémentée et un nouveau solveur BEM est obtenu. L'élimination des fréquences irrégulières est incluse. Le nouveau solveur est validé par comparaison des coefficients hydrodynamiques à des solutions analytiques pour une hémisphère, ainsi qu'à des résultats numériques obtenus avec un solveur commercial pour un chaland parallèlépipédique et le porte-conteneurs KCS. / The boundary element method (BEM) with constant panels is a common approach for wave-structure interaction problems. It is based on the linear potential-flow theory. It relies on the frequency-domain free-surface Green function, which is the focus of this thesis. First, the mathematical expressions and numerical methods for the frequency-domain free-surface Green function are investigated. Twelve different expressions are reviewed and analyzed. Several existing numerical methods are compared including their computational time and accuracies. Then, a series of ordinary differential equations (ODEs) for the time-domain and frequency-domain free-surface Green functions and their derivatives are derived. These ODEs can be used to better understand the properties of the Green function and can be an alternative way to calculate the Green functions and their derivatives. However, it is challenging to solve the ODEs for the frequency-domain Green function with initial conditions at the origin due to the singularity. This difficulty is removed by modifying the ODEs by using new functions free of singularity. The new ODEs are then transformed in their canonic form by using a novel definition of the vector functions. The canonic form can be solved with the initial conditions at the origin since all involved terms are finite. An expansion method based on series of logarithmic function together with ordinary polynomials which is very efficient for low frequency problems is also developed to obtain analytical solutions. Finally, the ODE-based method to calculate the Green function is implemented and an efficient BEM solver is obtained. The removal of irregular frequencies is included. The new solver is validated by comparison of hydrodynamic coefficients to analytical solutions for a heaving and surging hemisphere, and to numerical results obtained with a commercial solver for a box barge and the KCS container ship.

L’éolien offshore est l’énergie marine la plus avancée et utilisée dans le monde. Afin d’accroître l’énergie extraite du vent, les dimensions des éoliennes deviennent plus importantes et les parcs éoliens sont installées de plus en plus loin des côtes, où les mers sont plus agitées et les vents plus forts. De fait, les opérations marines sont plus complexes et plus chères et les fenêtres météo sont écourtées et se raréfient. Dans le cadre de cette thèse, un logiciel de simulation numérique des opérations marines est développé, en particulier pour des applications de descentes et de remontées de colis lourds. L’Algorithme aux Corps Rigides Composites, implémenté dans le logiciel InWave, est utilisé pour modéliser le système multicorps. Un modèle de câble et de treuil est développé, suivant la théorie multicorps utilisée, et comparé à la théorie câble classique dite « lumped mass ». Les efforts hydrodynamiques ainsi que les interactions hydrodynamiques sont modélisés par une théorie potentiel instationnaire satisfaisant l’hypothèse de faible perturbation, dite « weak-scatterer ». L’approche « weak-scatterer » du logiciel WS_CN est étendue aux simulations multi-flotteurs et validée par comparaison avec des données expérimentales. InWave et WS_CN sont couplés afin de résoudre l’interaction houle-structure pour des systèmes multicorps articulés en mer. Un couplage fort est adopté pour sa robustesse. L’équation de couplage est établie et validée via des comparaisons avec WS_CN. Le logiciel ainsi crée se nomme InWaveS_CN et utilise un code d’intégration en Python. Une nouvelle stratégie de maillage, basée sur un algorithme de découpe de maillages et une méthode par avance de front, est développée dans WS_CN. Enfin, des essais en bassin d’une opération de redressement ont été menés à l’ECN. La comparaison entre les simulations numériques et les données expérimentales offre une première et prometteuse validation d’InWaveS_CN. / Offshore wind represents the most advanced and used marine energy in the world. To increase the wind power extraction, turbines grow in size and wind farms are installed further offshore in presence of rough seas and strong winds. Marine operations become more challenging and expensive, weather windows are shorter and less frequent. This PhD work focuses on the development of a numerical tool to simulate marine operations with consistency, in particular lowering and lifting operations. The Composite-Rigid-Body Algorithm, implemented in the numerical tool InWave, is used to model multibody systems. A cable model and a winch model are developed following this multibody approach and compared to the classical low-order lumped mass theory. Hydrodynamic loads and hydrodynamic interactions are simulated using an unsteady potential flow theory based on the weakscatterer hypothesis, implemented in the numerical tool WS_CN. This approach is extended to multibody simulations and validated with comparisons to experimental data. InWave and WS_CN are coupled to solve wavestructure interaction for articulated multibody systems with large relative motions in waves. A tight coupling is selected for its robustness. The coupling equation is derived and validated from comparisons with WS_CN. This leads to the creation of a new numerical tool, InWaveS_CN, using Python as glue code language. A new mesh strategy, based on the coupling between a panel cutting method and an advance front method, is developed in WS_CN. Experiments of an upending operation were conducted at Ecole Centrale de Nantes. The comparison between the numerical simulations and the experimental data leads to a first and promising validation of InWaveS_CN.

No description available.

Ocean wave energy is a promising renewable source to contribute to supplying the world’s energy demand. The Division of Electricity at Uppsala University is developing a technology to capture energy from ocean waves with a wave energy converter (WEC) consisting of a linear permanent magnet generator and a point absorber. The linear generator is placed on sea bed and is driven directly by the floating absorber. Since March 2006, multiple wave energy converters have been deployed on the Swedish west coast outside the town of Lysekil. The technology is verified by long-term operation during at sea and satisfactory reliability of the electricity generation. This thesis focuses on developing advanced control strategies for fully coupled wave energy converters subject to constraints. A nonlinear control strategy is studied in detail for a single WEC subject to constraints under regular and irregular waves. Besides, two coordinated control strategies are developed to investigate the performance of a wave energy farm subject to constraints. The performance of the WECs using these control strategies are investigated in case studies, and optimal PTO damping coefficients are found to maximize the output power. The results show that these control strategies can significantly improve the performance of the WECs, in terms of mean power, compared to a conventional control. Besides these control strategies, a wave-to-wire simulation platform is built to study the power generation control of the WEC subject to constraints.  The wave-to-wire simulation platform allows both nonlinear and linear control force. The results show that there is a good agreement between the desired value and the actual value after advanced control.

Em geral, uma embarcação de planeio é projetada para atingir elevados níveis de velocidade. Esse atributo de desempenho está diretamente relacionado ao porte da embarcação e à potência instalada em sua planta propulsiva. Tradicionalmente, durante o projeto de uma embarcação, as análises de desempenho são realizadas através de resultados de embarcações já existentes, retirados de séries sistemáticas ou de embarcações já desenvolvidas pelo estaleiro e/ou projetista. Além disso, a determinação dos atributos de desempenho pode ser feita através de métodos empíricos e/ou estatísticos, onde a embarcação é representada através de seus parâmetros geométricos principais; ou a partir de testes em modelos em escala reduzida ou protótipos. No caso específico de embarcações de planeio, o custo dos testes em escala reduzida é muito elevado em relação ao custo de projeto. Isso faz com que a maioria dos projetistas não opte por ensaios experimentais das novas embarcações em desenvolvimento. Ao longo dos últimos anos, o método de Savitsky foi largamente utilizado para se realizar estimativas de potência instalada de uma embarcação de planeio. Esse método utiliza um conjunto de equações semi-empíricas para determinar os esforços atuantes na embarcação, a partir dos quais é possível determinar a posição de equilíbrio de operação e a força propulsora necessária para navegar em uma dada velocidade. O método de Savitsky é muito utilizado nas fases iniciais de projeto, onde a geometria do casco ainda não foi totalmente definida, pois utiliza apenas as características geométricas principais da embarcação para realização das estimativas de esforços. À medida que se avança nas etapas de projeto, aumenta o detalhamento necessário das estimativas de desempenho. Para a realização, por exemplo, do projeto estrutural é necessária uma estimativa do campo de pressão atuante no fundo do casco, o qual não pode ser determinado pelo método de Savitsky. O método computacional implementado nesta dissertação, tem o objetivo de determinar as características do escoamento e o campo de pressão atuante no casco de uma embarcação de planeio navegando em águas calmas. O escoamento é determinado através de um problema de valor de contorno, no qual a superfície molhada no casco é considerada um corpo esbelto. Devido ao uso da teoria de corpo esbelto o problema pode ser tratado, separadamente, em cada seção, onde as condições de contorno são forçadamente respeitadas através de uma distribuição de vórtices. / Generally, a planing craft is designed to achieve high speed levels. This performance attribute is directly related to the boat size and to the propeller plant power. Traditionally, during a boat design, performance analyses are carried out using results taken from systematic series or from others boat previously build by the shipyard and/or designer. Furthermore, performance attributes can be calculated by semi-empirical and/or statistic methods or by tests of reduced scale models. In the specific case of planing boats, the costs of reduced scale tests are too high compared to the design cost itself. Because of this, most designers do not perform experimental tests during the development of new boats. During the last years, the Savitsky method was extensively used to estimate planing craft effective power. The method uses a set of semi-empirical equations to calculate the forces acting on the boat, from which the equilibrium position and the required propeller thrust are determined. During the preliminary phases of planing craft design, the hull geometry hasn\'t been fully defined. Therefore, the Savitsky method is widely used during this phase, because it uses only the main geometrical characteristics to estimate the forces acting on the hull. Advancing toward the final phases of the design process, more detailed information is required. To execute the structural design, for example, the pressure field acting on the hull must be known, which can\'t be estimate using the Savitsky method. The main objective of the present study is to implement a computer method that can be used to estimate the fluid flow and pressure field acting on the hull of a boat moving with forward speed constant in calm water. The fluid flow around the hull is treated as a boundary value problem, in which the wetted hull surface is considered a slender body. The slender body theory enables to solve the problem separately, in each transverse section, where boundary conditions are respected by a sheet of vortices.

Em geral, uma embarcação de planeio é projetada para atingir elevados níveis de velocidade. Esse atributo de desempenho está diretamente relacionado ao porte da embarcação e à potência instalada em sua planta propulsiva. Tradicionalmente, durante o projeto de uma embarcação, as análises de desempenho são realizadas através de resultados de embarcações já existentes, retirados de séries sistemáticas ou de embarcações já desenvolvidas pelo estaleiro e/ou projetista. Além disso, a determinação dos atributos de desempenho pode ser feita através de métodos empíricos e/ou estatísticos, onde a embarcação é representada através de seus parâmetros geométricos principais; ou a partir de testes em modelos em escala reduzida ou protótipos. No caso específico de embarcações de planeio, o custo dos testes em escala reduzida é muito elevado em relação ao custo de projeto. Isso faz com que a maioria dos projetistas não opte por ensaios experimentais das novas embarcações em desenvolvimento. Ao longo dos últimos anos, o método de Savitsky foi largamente utilizado para se realizar estimativas de potência instalada de uma embarcação de planeio. Esse método utiliza um conjunto de equações semi-empíricas para determinar os esforços atuantes na embarcação, a partir dos quais é possível determinar a posição de equilíbrio de operação e a força propulsora necessária para navegar em uma dada velocidade. O método de Savitsky é muito utilizado nas fases iniciais de projeto, onde a geometria do casco ainda não foi totalmente definida, pois utiliza apenas as características geométricas principais da embarcação para realização das estimativas de esforços. À medida que se avança nas etapas de projeto, aumenta o detalhamento necessário das estimativas de desempenho. Para a realização, por exemplo, do projeto estrutural é necessária uma estimativa do campo de pressão atuante no fundo do casco, o qual não pode ser determinado pelo método de Savitsky. O método computacional implementado nesta dissertação, tem o objetivo de determinar as características do escoamento e o campo de pressão atuante no casco de uma embarcação de planeio navegando em águas calmas. O escoamento é determinado através de um problema de valor de contorno, no qual a superfície molhada no casco é considerada um corpo esbelto. Devido ao uso da teoria de corpo esbelto o problema pode ser tratado, separadamente, em cada seção, onde as condições de contorno são forçadamente respeitadas através de uma distribuição de vórtices. / Generally, a planing craft is designed to achieve high speed levels. This performance attribute is directly related to the boat size and to the propeller plant power. Traditionally, during a boat design, performance analyses are carried out using results taken from systematic series or from others boat previously build by the shipyard and/or designer. Furthermore, performance attributes can be calculated by semi-empirical and/or statistic methods or by tests of reduced scale models. In the specific case of planing boats, the costs of reduced scale tests are too high compared to the design cost itself. Because of this, most designers do not perform experimental tests during the development of new boats. During the last years, the Savitsky method was extensively used to estimate planing craft effective power. The method uses a set of semi-empirical equations to calculate the forces acting on the boat, from which the equilibrium position and the required propeller thrust are determined. During the preliminary phases of planing craft design, the hull geometry hasn\'t been fully defined. Therefore, the Savitsky method is widely used during this phase, because it uses only the main geometrical characteristics to estimate the forces acting on the hull. Advancing toward the final phases of the design process, more detailed information is required. To execute the structural design, for example, the pressure field acting on the hull must be known, which can\'t be estimate using the Savitsky method. The main objective of the present study is to implement a computer method that can be used to estimate the fluid flow and pressure field acting on the hull of a boat moving with forward speed constant in calm water. The fluid flow around the hull is treated as a boundary value problem, in which the wetted hull surface is considered a slender body. The slender body theory enables to solve the problem separately, in each transverse section, where boundary conditions are respected by a sheet of vortices.

The master's thesis deals with Navier-Stokes equations in curvilinear coordinates and their solution for quasi-potential flow. The emphasis is on detailed description of curvilinear space and its expression using Bézier curves, Bézier surfaces and Bézier bodies. Further, fundamental concepts of hydromechanics are defined, including potential and quasi-potential flow. Cauchy equations are derived as a result of the law of momentum conservation and continuity equation is derived as a result of principle of mass conservation. Navier-Stokes equations are then derived as a special case of Cauchy equations using Cauchy stress tensor of Newtonian compressible fluid. Further transformation into curvilinear coordinates is accomplished through differential operators in curvilinear coordinates and by using curvature vector of space curve. In the last section we use results from previous chapters to solve boundary value problem of quasi-potential flow, which was solved by finite difference method using Matlab environment.

En aquest treball s'ha estudiat la dinàmica dels làsers de classe A i de classe B en termes del potencial de Lyapunov. En el cas que s'injecti un senyal al làser o es modulin alguns dels paràmetres, apareix un comportament moltmés complex i s'estudia el conjunt de bifurcacions.1) Als làsers de classe A, la dinàmica determinista s'ha interpretat com el moviment damunt el potencial de Lyapunov. En la dinàmica estocàstica s'obté un flux sostingut per renou per a la fase del camp elèctric.2) Per als làsers de classe A amb senyal injectat, s'ha descrit el conjunt de bifurcacions complet i s'ha determinat el conjunt d'amplituds i freqüències en el quals el làser responajustant la seva freqüència a la del camp extern. 3) S'ha obtingut un potencial de Lyapunov pels làsers de classe B, només vàlid en el cas determinista, que inclou els termes de saturació de guany i d'emissió espontània.4) S'ha realitzat un estudi del conjunt de bifurcacions parcial al voltant del règim tipus II de la singularitat Hopf--sella--node en un làser de classe B amb senyal injectat.5) S'han identificat les respostes òptimes pels làsers de semiconductor sotmesos a modulació periòdica externa. S'han obtingut les corbes que donen la resposta màxima per cada tipus de resonància en el pla definit per l'amplitud relativa de modulació i la freqüència de modulació. / In this work we have studied the dynamics of both class A and class B lasers in terms of Lyapunov potentials. In the case of an injected signal or when some laser parameters are modulated, and more complex behaviour is expected, the bifurcation set is studied. The main results are the following:1) For class A lasers, the deterministic dynamics has been interpreted as a movement on the potential landscape. In the stochastic dynamics we have found a noise sustained flow for the phase of the electric field. 2) For class A lasers with an injected signal, we have been able to describe the whole bifurcation set of this system and to determine the set of amplitudes frequencies for which the laser responds adjusting its frequency to that of the external field. 3) In the case of class B lasers, we have obtained a Lyapunov potential only valid in the deterministic case, including spontaneous emission and gain saturation terms. The fixed point corresponding to the laser in the on state has been interpreted as a minimum in this potential. Relaxation to this minimum is reached through damped oscillations. 4) We have performed a study of the partial bifurcation set around the type II regime of the Hopf-saddle-node singularity in a class B laser with injected signal. 5) We have identified the optimal responses of a semiconductor laser subjected to an external periodic modulation. The lines that give a maximum response for each type of resonance are obtained in the plane defined by the relative amplitude modulation and frequency modulation.