Abstract: A computer-readable medium stores instructions including storing a model of a physical structure and defining a mesh for the model. The mesh includes finite elements, each defined by a respective set of edges. The instructions include, for each finite element, identifying a governing differential equation and a set of complementary functions that exactly satisfy the governing differential equation. The instructions include determining an applied physical stimulus for the physical structure. The instructions include generating an energy optimization model that minimizes a difference between internal energy and external energy of the finite elements in response to the applied physical stimulus. The model includes a matrix of respective scalar multipliers for the complementary functions for each finite element. The instructions include transforming the matrix and calculating a physical parameter of interest.

Abstract: A method of evaluating response of a physical structure to external stimulus includes defining a mesh of finite elements, each defined by edges, for a model of the physical structure. The method includes identifying a governing differential equation and associated complementary functions, which are each associated with a scalar multiplier. The method includes generating an energy optimization model that minimizes a difference between internal and external energies of the finite elements. The internal energy is based on strain energy in a volume defined by the edges of the finite element and resulting from deformations by the complementary functions. The external energy of each finite element is based on external work done by the external stimulus acting on the finite element as deformed by the complementary functions. The method includes solving the energy optimization model for the scalar multipliers and calculating a resulting parameter of interest of the physical structure.

Abstract: A method of evaluating response of a physical structure to external stimulus includes defining a mesh of finite elements, each defined by edges, for a model of the physical structure. The method includes identifying a governing differential equation and associated complementary functions, which are each associated with a scalar multiplier. The method includes generating an energy optimization model that minimizes a difference between internal and external energies of the finite elements. The internal energy is based on strain energy in a volume defined by the edges of the finite element and resulting from deformations by the complementary functions. The external energy of each finite element is based on external work done by the external stimulus acting on the finite element as deformed by the complementary functions. The method includes solving the energy optimization model for the scalar multipliers and calculating a resulting parameter of interest of the physical structure.

Abstract: A method of evaluating response of a physical structure to external stimulus includes defining a mesh of finite elements, each defined by edges, for a model of the physical structure. The method includes identifying a governing differential equation and associated complementary functions, which are each associated with a scalar multiplier. The method includes generating an energy optimization model that minimizes a difference between internal and external energies of the finite elements. The internal energy is based on strain energy in a volume defined by the edges of the finite element and resulting from deformations by the complementary functions. The external energy of each finite element is based on external work done by the external stimulus acting on the finite element as deformed by the complementary functions. The method includes solving the energy optimization model for the scalar multipliers and calculating a resulting parameter of interest of the physical structure.

Abstract: Methods, systems, and computer-readable media generate acceleration time histories. An initial acceleration history is applied to a response model with natural frequencies across a spectrum of interest to develop a displacement response. Low-frequency enhancement signals are determined by comparing the displacement response to a standard displacement response. The enhancement signals are combined with the initial acceleration history to develop a second acceleration history, which is applied to the response model to develop an acceleration response. High-frequency enhancement signals are determined by comparing the acceleration response to a standard acceleration response. The enhancement signals are combined with the second acceleration history to develop a desired acceleration history.

Abstract: Our invention relates to the feeding of sodium to aluminum containing silicon as an alloying element in hypoeutectic percentages by flowing the molten aluminum through a trough and feeding a tube which contains sodium within it into the aluminum in the trough, the tube being preferably fed in countercurrent direction with special mixing and a protective surface coating for the flowing molten aluminum. The invention appears primarily applicable to casting large masses of aluminum such as are used in the melts which produce aluminum ingots for sale to foundries.