Flugge was born on 16 July 1903 in Hameln, Germany. She graduated from high school in 1923 and entered the male-dominated Technical University in Hanover, studying applied mathematics. She was frequently the only woman in her classes. She earned the degree of Diplom-Ingenieur, and a Doktor-Ingenieur in 1929. During her studies she was a teaching assistant.
She took up work with the Aerodyamische Versuchsanstalt in Gottingen, a research institute. In 1938 her married Dr. Wilhelm Flugge, a civil engineer. In the mid-1940s they moved to Paris to work at the French National Office of Aeronautical Research. In 1948 both were offered positions at Stanford University, in Palo Alto, CA. Irmgard was the first woman at Stanford to be named a full Professor of Engineering. Her research would involve fluid dynamics, finite difference methods and the use of computers. In her latter years she did work in automatic control theory.
She was a Senior Member of the IEEE.
She retired in 1968 and passed away on 22 May 1974.